Directory UMM :Journals:Journal_of_mathematics:OTHER:
71
Documenta Math.
Motivic Tubular Neighborhoods
Marc Levine
Received: November 12, 2005
Revised: January 2, 2007
Communicated by Alexander Merkurjev
Abstract. We construct motivic versions of the classical tubular
neighborhood and the punctured tubular neighborhood, and give applications to the construction of tangential base-points for mixed Tate
motives, algebraic gluing of curves with boundary components, and
limit motives.
2000 Mathematics Subject Classification: Primary 14C25; Secondary
55P42, 18F20, 14F42
Keywords and Phrases: Limit motives, moduli of curves, Tate motives
1. Introduction
Let i : A → B be a closed embedding of finite CW complexes. One useful
fact is that A admits a cofinal system of neighborhoods T in B with A → T a
deformation retract. This is often used in the case where B is a differentiable
manifold, showing for example that A has the homotopy type of the differentiable manifold T . This situation occurs in algebraic geometry, for instance
in the case of the inclusion of the special fiber in a degeneration of smooth
varieties X → C over the complex numbers.
To some extent, one has been able to mimic this construction in purely algebraic
terms. The rigidity theorems of Gillet-Thomason [14], extended by Gabber
(details appearing a paper of Fujiwara [13]) indicated that, at least through
the eyes of torsion ´etale sheaves, the topological tubular neighborhood can be
replaced by the Hensel neighborhood. However, basic examples of non-torsion
phenomena, even in the ´etale topology, show that the Hensel neighborhood
cannot always be thought of as a tubular neighborhood, perhaps the simplest
example being the sheaf Gm .1
Our object in this paper is to construct an algebraic version of the tubular
neighborhood which has the basic properties of the topological construction,
at least for a reasonably large class of cohomology theories. It turns out that
1If O is a local ring with residue field k and maximal ideal m, the surjection G (O) →
m
Gm (k) has kernel (1 + m)× , which is in general non-zero, even for O Hensel
Documenta Mathematica 12 (2007) 71–146
72
Marc Levine
a “homotopy invariant” version of the Hensel neighborhood does the job, at
least for theories which are themselves homotopy invariant. If one requires in
addition that the given cohomology theory has a Mayer-Vietoris property for
the Nisnevich topology, then one also has an algebraic version of the punctured
tubular neighborhood. We extend these constructions to the case of a (reduced)
strict normal crossing subscheme by a Mayer-Vietoris procedure, giving us
the tubular neighborhood and punctured tubular neighborhood of a normal
crossing subscheme of a smooth k-scheme.
Morel and Voevodsky have constructed an algebro-geometric version of homotopy theory, in the setting of presheaves of spaces or spectra on the category
of smooth varieties over a reasonable base scheme B; we concentrate on the
A1 -homotopy category of spectra, SHA1 (B). For a map f : X → Y , they
construct a pair of adjoint functors
Rf∗ : SHA1 (X) → SHA1 (Y )
Lf ∗ : SHA1 (Y ) → SHA1 (X).
If we have a closed immersion i : W → X with open complement j : U → X,
then one has the functor
Li∗ Rj∗ : SHA1 (U ) → SHA1 (W )
One of our main results is that, in case W is a strict normal crossing subscheme
of a smooth X, the restriction of Li∗ Rj∗ E to a Zariski presheaf on W can be
viewed as the evaluation of E on the punctured tubular neighborhood of W in
X.
Consider a morphism p : X → A1 and take i : W → X to be the inclusion
of p−1 (0). Following earlier constructions of Spitzweck [43], Ayoub has constructed a “unipotent specialization functor” in the motivic setting, essentially
(in the case of a semi-stable degeneration) by evaluating Li∗ Rj∗ E on a cosimplicial version of the appropriate path space on Gm with base-point 1. Applying
the same idea to our tubular neighborhood construction gives a model for this
specialization functor, again only as a Zariski presheaf on p−1 (0).
Ayoub has also defined a motivic monodromy operator and monodromy sequence involving the unipotent specialization functor and the functor Li∗ Rj∗ ,
for theories with Q-coefficients that satisfy a certain additional condition (see
definition 9.2.2). We give a model for this construction by combining our
punctured tubular neighborhood with a Q-linear version of the Gm -path space
mentioned above. We conclude with an application of our constructions to the
moduli spaces of smooth curves and a construction of a specialization functor for category of mixed Tate motives, which in some cases yields a purely
algebraic construction of tangential base-points. Of course, the construction
of Ayoub, when restricted to the triangulated category of Tate motives, also
gives such a specialization functor, but we hope that the explicit nature of our
construction will be useful for applications.
We have left to another paper the task of checking the compatibilities of our
constructions with others via the appropriate realization functor. As we have
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
73
already mentioned, our punctured tubular neighborhood construction is comparable with the motivic version of the functor Li∗ Rj∗ for the situation of a
normal crossing scheme i : D → X with open complement j : X \ D → X;
this should imply that it is a model for the analogous functor after realization.
Similarly, our limit cohomology construction should transform after realization
to the appropriate version of the sheaf of vanishing cycles, at least in the case of
a semi-stable degeneration, and should be comparable with the constructions
of Rappaport-Zink [37] as well as the limit mixed Hodge structure of Katz [22]
and Steenbrink [44]. Our specialization functor for Tate motives should be
compatible with the Betti, ´etale and Hodge realizations; similarly, realization
functors applied to our limit motive should yield for example the limit mixed
Hodge structure. We hope that our rather explicit construction of the limiting
motive will be useful in giving a geometric view to the limit mixed Hodge structure of a semi-stable degeneration but we have not attempted an investigation
of these issues in this paper.
My interest in this topic began as a result of several discussions on limit motives with Spencer Bloch and H´el`ene Esnault, whom I would like to thank for
their encouragement and advice. I would also like to thank H´el`ene Esnault for
clarifying the role of the weight filtration leading to the exactness of ClemensSchmidt monodromy sequence (see Remark 9.3.6). An earlier version of our
constructions used an analytic (i.e. formal power series) neighborhood instead
of the Hensel version now employed; I am grateful to Fabien Morel for suggesting this improvement. Finally, I want to thank Joseph Ayoub for explaining
his construction of the nearby cycles functor; his comments suggested to us
the use of the cosimplicial path space in our construction of limit cohomology.
In addition, Ayoub noticed a serious error in our first attempt at constructing the monodromy sequence; the method used in this version is following his
suggestions and comments. Finally, we would like to thank the referee for giving unusually thorough and detailed comments and suggestions, which have
substantially improved this paper. In particular, the material in sections 7
comparing our construction with the categorical ones of Morel-Voevodsky, as
well as the comparison with Ayoub’s specialization functor and monodromy
sequence in section 8.3 and section 9 was added following the suggestion of the
referee, who also supplied the main ideas for the proofs.
2. Model structures and other preliminaries
2.1. Presheaves of simplicial sets. We recall some facts on the model
structures in categories of simplicial sets, spectra, associated presheaf categories
and certain localizations. For details, we refer the reader to [17] and[19].
For a small category I and category C, we will denote the category of functors
from I to C by C I .
We let Ord denote the category with objects the finite ordered sets [n] :=
{0, . . . , n} (with the standard ordering) and morphisms the order-preserving
op
maps of sets. For a category C, the functor categories C Ord , C Ord are the
Documenta Mathematica 12 (2007) 71–146
74
Marc Levine
categories of cosimplicial objects of C, resp. simplicial objects of C. For C =
op
Sets, we have the category of simplicial sets, Spc := SetsOrd , and similarly
for C the category of pointed sets, Sets∗ , the category of pointed simplicial sets
op
.
Spc∗ := SetsOrd
∗
We give Spc and Spc∗ the standard model structures: cofibrations are
(pointed) monomorphisms, weak equivalences are weak homotopy equivalences
on the geometric realization, and fibrations are detemined by the right lifting property (RLP) with respect to trivial cofibrations; the fibrations are then
exactly the Kan fibrations. We let |A| denote the geometric realization, and
[A, B] the homotopy classes of (pointed) maps |A| → |B|.
For an essentially small category C, we let Spc(C) be the category of presheaves
of simplicial sets on C. We give Spc(C) the so-called injective model structure,
that is, the cofibrations and weak equivalences are the pointwise ones, and the
fibrations are determined by the RLP with respect to trivial cofibrations. We
let HSpc(C) denote the associated homotopy category (see [17] for details on
these model structures for Spc and Spc(C)).
2.2. Presheaves of spectra. Let Spt denote the category of spectra. To
fix ideas, a spectrum will be a sequence of pointed simplicial sets E0 , E1 , . . .
together with maps of pointed simplicial sets ǫn : S 1 ∧ En → En+1 . Maps of
spectra are maps of the underlying simplicial sets which are compatible with
the attaching maps ǫn . The stable homotopy groups πns (E) are defined by
πns (E) := lim [S m+n , Em ].
m→∞
The category Spt has the following model structure: Cofibrations are maps
f : E → F such that E0 → F0 is a cofibration, and for each n ≥ 0, the map
a
S 1 ∧ Fn → Fn+1
En+1
S 1 ∧En
is a cofibration. Weak equivalences are the stable weak equivalences, i.e., maps
f : E → F which induce an isomorphism on πns for all n. Fibrations are
characterized by having the RLP with respect to trivial cofibrations. We write
SH for the homotopy category of Spt.
For X ∈ Spc∗ , we have the suspension spectrum Σ∞ X := (X, ΣX, Σ2 X, . . .)
with the identity bonding maps. Dually, for a spectrum E := (E0 , E1 , . . .) we
have the 0-space Ω∞ E := limn Ωn En . These operations form a Quillen pair
of adjoint functors (Σ∞ , Ω∞ ) between Spc∗ and Spt, and thus induce adjoint
functors on the homotopy categories.
Let C be a category. A functor E : C op → Spt is called a presheaf of spectra on
C.
We use the following model structure on the category of presheaves of spectra (see [19]): Cofibrations and weak equivalences are given pointwise, and
fibrations are characterized by having the RLP with respect to trivial cofibrations. We denote this model category by Spt(C), and the associated homotopy
category by HSpt(C).
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
75
As a particular example, we have the model category of simplicial spectra
op
SptOrd = Spt(Ord). We have the total spectrum functor
Tot : Spt(Ord) → Spt
which preserves weak equivalences. The adjoint pair (Σ∞ , Ω∞ ) extend pointwise to define a Quillen pair on the presheaf categories and an adjoint pair on
the homotopy categories.
Let B be a noetherian separated scheme of finite Krull dimension. We let
Sm/B denote the category of smooth B-schemes of finite type over B. We
often write Spc(B) and HSpc(B) for Spc(Sm/B) and HSpc(Sm/B), and
write Spt(B) and HSpt(B) for Spt(Sm/B) and HSpt(Sm/B).
For Y ∈ Sm/B, a subscheme U ⊂ Y of the form Y \ ∪α Fα , with {Fα } a
possibly infinite set of closed subsets of Y , is called essentially smooth over B;
the category of essentially smooth B-schemes is denoted Smess .
2.3. Local model structure. If the category C has a topology, there is often
another model structure on Spc(C) or Spt(C) which takes this into account. We
consider the case of the small Nisnevich site XNis on a scheme X (assumed to
be noetherian, separated and of finite Krull dimension), and the big Nisnevich
sites Sm/BNis or Sch/BNis , as well as the Zariski versions XZar , Sm/BZar ,
etc. We describe the Nisnevich version for spectra below; the definitions and
results for the Zariski topology and for spaces are exactly parallel.
Definition 2.3.1. A map f : E → F of presheaves of spectra on XNis is a local
weak equivalence if the induced map on the Nisnevich sheaf of stable homotopy
s
s
(F )Nis is an isomorphism of sheaves for all m. A
(E)Nis → πm
groups f∗ : πm
map f : E → F of presheaves of spectra on Sm/BNis or Sch/BNis is a local
weak equivalence if the restriction of f to XNis is a local weak equivalence for
all X ∈ Sm/B or X ∈ Sch/B.
The Nisnevich local model structure on the category of presheaves of spectra
on XNis has cofibrations given pointwise, weak equivalences the local weak
equivalences and fibrations are characterized by having the RLP with respect to trivial cofibrations. We write Spt(XNis ) for this model category, and
HSpt(XNis ) for the associated homotopy category. The Nisnevich local model
categories Spt(Sm/BNis ) and Spt(Sch/BNis ), with homotopy categories
HSpt(Sm/BNis ) and HSpt(Sch/BNis ), are defined similarly. A similar localization gives model categories of presheaves of spaces Spc(XNis ), Spc(XZar ),
Spc(Sm/BNis ), etc., and homotopy categories HSpc(XNis ), HSpc(XZar ),
HSpc(Sm/BNis ), etc. We also have the adjoint pair (Σ∞ , Ω∞ ) in this setting. For details, we refer the reader to [19].
Documenta Mathematica 12 (2007) 71–146
76
Marc Levine
Remark 2.3.2. Let E be in Spt(Sm/BNis ), and let
W′
i′
/ Y′
f
W
i
/X
be an elementary Nisnevich square, i.e., f is ´etale, i : W → X is a closed
immersion, the square is cartesian, and W ′ → W is an isomorphism, with X
and X ′ in Sm/B (see [34, Definition 1.3, pg. 96]) .
If E is fibrant in Spt(Sm/BNis ) then E transforms each elementary Nisnevich
square to a homotopy cartesian square in Spt. Conversely, suppose that E
transforms each elementary Nisnevich square to a homotopy cartesian square
in Spt. Then E is quasi-fibrant, i.e., for all Y ∈ Sm/B, the canonical map
E(Y ) → Ef ib (Y ), where Ef ib is the fibrant model of E, is a weak equivalence.
See [19] for details.
If we define an elementary Zariski square as above, with X ′ → X an open
immersion, the same holds in the model category Spt(Sm/BZar ). More precisely, one can show (see e.g. [45]) that, if E transforms each elementary Zarisk
square to a homotopy cartesian square in Spt, then E satisfies Mayer-Vietoris
for the Zariski topology: if X ∈ Sm/B is a union of Zariski open subschemes
U and V , then the evident sequence
E(X) → E(U ) ⊕ E(V ) → E(U ∩ V )
is a homotopy fiber sequence in SH.
Remark 2.3.3. Let C be a small category with an initial object ∅ and admiting
finite coproducts over ∅, denoted X ∐ Y . A functor E : C op → Spt is called
additive if for each X, Y in C, the canonical map
E(X ∐ Y ) → E(X) ⊕ E(Y )
in SH is an isomorphism. It is easy to show that if E ∈ Spt(Sm/B) satisfies
Mayer-Vietoris for the Zariski topology, and E(∅) ∼
= 0 in SH, then E is additive.
From now on, we will assume that all our presheaves of spectra E satisfy
E(∅) ∼
= 0 in SH.
2.4. A1 -local structure. One can perform a Bousfield localization on
Spc(Sm/BNis ) or Spt(Sm/BNis ) so that the maps Σ∞ X × A1+ → Σ∞ X+
induced by the projections X × A1 → X become weak equivalences. We call
the resulting model structure the Nisnevich-local A1 -model structure, denoted
SpcA1 (Sm/BNis ) or SptA1 (Sm/BNis ). One has the Zariski-local versions as
well. We denote the homotopy categories for the Nisnevich version by HA1 (B)
(for spaces) and SHA1 (B) (for spectra). For the Zariski versions, we indicate
the topology in the notation. We also have the adjoint pair (Σ∞ , Ω∞ ) in this
setting. For details, see [30, 31, 34].
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
77
2.5. Additional notation. Given W ∈ Sm/S, we have restriction functors
Spc(S) → Spc(WZar )
Spt(S) → Spt(WZar );
we write the restriction of some E ∈ Spc(S) to Spc(WZar ) as E(WZar ). We
use a similar notation for the restriction of E to Spt(WZar ), or for restrictions
to WNis . More generally, if p : Y → W is a morphism in Sm/S, we write
E(Y /WZar ) for the presheaf U 7→ E(Y ×W U ) on WZar .
For Z ⊂ Y a closed subset, Y ∈ Sm/S and for E ∈ Spc(S) or E ∈ Spt(S),
we write E Z (Y ) for the homotopy fiber of the restriction map
E(Y ) → E(Y \ Z).
ZZar
We define the presheaf E
(Y ) by setting, for U ⊂ Z a Zariski open subscheme with closed complement F ,
E ZZar (Y )(U ) := E U (Y \ F ).
A co-presheaf on a category C with values in A is just an A-valued preheaf on
C op .
As usual, we let ∆n denote the algebraic n-simplex
X
∆n := Spec Z[t0 , . . . , tn ]/
ti − 1,
i
and ∆∗ the cosimplicial scheme n 7→ ∆n . For a scheme X, we have ∆nX :=
X × ∆n and the cosimplicial scheme ∆∗X .
Let B be a scheme as above. For E ∈ Spc(B) or in Spt(B), we say that E is
homotopy invariant if for all X ∈ Sm/B, the pull-back map E(X) → E(X×A1 )
is a weak equivalence (resp., stable weak equivalence). We say that E satisfies
Nisnevich excision if E transforms elementary Nisnevich squares to homotopy
cartesian squares.
3. Tubular neighborhoods for smooth pairs
Let i : W → X be a closed immersion in Sm/k. In this section, we construct the
ˆ
tubular neighborhood τǫX (W ) of W in X as a functor from WZar to cosimplicial
ˆ
pro-k-schemes. Given E ∈ Spc(k), we can evaluate E on τǫX (W ), yielding the
ˆ
X
presheaf of spaces E(τǫ (W )) on WZar , which is our main object of study.
ˆ
3.1. The cosimplicial pro-scheme τǫX (W ). For a closed immersion W → T
W
in Sm/k, let TNis
be the category of Nisnevich neighborhoods of W in T , i.e.,
objects are ´etale maps p : T ′ → T of finite type, together with a section
s : W → T ′ to p over W . Morphisms are morphisms over T which respect the
W
sections. Note that TNis
is a left-filtering essentially small category.
′
h
Sending (p : T → T, s : W → T ′ ) to T ′ ∈ Sm/k defines the pro-object TˆW
of
′
Sm/k; the sections s : W → T give rise to a map of the constant pro-scheme
h
W to TˆW
, denoted
h
ˆiW : W → TˆW
.
Documenta Mathematica 12 (2007) 71–146
78
Marc Levine
Given a k-morphism f : S → T , and closed immersions iV : V → S, iW : W →
T such that f ◦ iV factors through iW (by f¯ : V → W ), we have the pull-back
functor
W
V
f ∗ : TNis
→ SNis
,
f ∗ (T ′ → T, s : W → T ′ ) := (T ′ ×T S, (s ◦ f¯, iV )).
h
This gives us the map of pro-objects f h : SˆVh → TˆW
, so that sending W → T
h
h
to TˆW and f to f becomes a pseudo-functor.
h
We let f h : SˆVh → TˆW
denote the induced map on pro-schemes. If f happens
to be a Nisnevich neighborhood of W → X (so f¯ : V → W is an isomorphism)
h
then f h : SˆVh → TˆW
is clearly an isomorphism.
h
Remark 3.1.1. The pseudo-functor (W → T ) 7→ TˆW
can be rectified to an
W
W
honest functor by first replacing TNis with the cofinal subcategory TNis,0
of
neighborhoods T ′ → T , s : W → T ′ such that each connected component of T ′
W
has only identity
has non-empty intersection with s(W ). One notes that TNis,0
W
W
automorphisms, so we replace TNis,0 with a choice of a full subcategory TNis,00
W
giving a set of representatives of the isomorphism classes in TNis,0 . Given a
i
i
W
V
→ T ) as above,
S) → (W −−
map of pairs of closed immersions f : (V −→
∗
we modify the pull-back functor f defined above by passing to the connected
component of (s ◦ f¯, iV )(V ) in T ′ ×T S. We thus have the honest functor
W
h
(W → T ) 7→ TNis,00
which yields an equivalent pro-object TˆW
.
As pointed out by the referee, one can also achieve strict functoriality by rectifying the fiber product; in any case, we will use a strictly functorial version
from now on without comment.
n )h . The
ˆn
d
For a closed immersion i : W → X in Sm/k, set ∆
:= (∆
n
X,W
cosimplicial scheme
X ∆W
∆∗X : Ord → Sm/k
[n] 7→ ∆nX
thus gives rise to the cosimplicial pro-scheme
ˆ∗
∆
: Ord → Pro-Sm/k
X,W
ˆn
[n] 7→ ∆
X,W
n )h
d
give the closed immersion of cosimplicial
The maps ˆi∆nW : ∆nW → (∆
X ∆n
W
pro-schemes
ˆ ∗X,W .
ˆiW : ∆∗W → ∆
ˆn
Also, the canonical maps πn : ∆
→ ∆n define the map
X,W
πX,W :
ˆ∗
∆
X,W
X
→ ∆∗X .
Let (p : X ′ → X, s : W → X ′ ) be a Nisnevich neighborhood of (W, X). The
map
ˆn
ˆn′ → ∆
p:∆
X,W
X ,W
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
79
is an isomorphism respecting the closed immersions ˆiW . Thus, sending a Zariski
ˆn
open subscheme U ⊂ W with complement F ⊂ W ⊂ X to ∆
X\F,U defines a
n
ˆ
on WZar with values in pro-objects of Sm/k; we write
co-presheaf ∆
ˆ
ˆ
X,W
Zar
τǫX (W ) for the cosimplicial object
ˆ nˆ
n 7→ ∆
X,W
Zar
.
ˆ in the notation because the co-presheaf ∆
ˆn
is depends only
We use the X
ˆ
X,W
Zar
on the Nisnevich neighborhood of W in X.
Let ∆∗WZar denote the co-presheaf on WZar defined by U 7→ ∆∗U . The closed
immersions ˆiU define the natural transformation
ˆiW : ∆∗W → τǫXˆ (W ).
Zar
The maps πX\F,W \F for F ⊂ W a Zariski closed subset define the map
ˆ
πX,W : τǫX (W ) → ∆∗X|WZar
where X|WZar is the co-presheaf W \ F 7→ X \ F on WZar . We let
(3.1.1)
ˆ
π
¯X,W : τǫX (W ) → X|WZar
denote the composition of πX.W with the projection ∆∗X|WZar → X|WZar .
3.2. Evaluation on spaces. Let i : W → T be a closed immersion in Sm/k.
For E ∈ Spc(T ), we have the space E(TˆhW ), defined by
h
E(TˆW
) :=
colim
W
(p:T ′ →T,s:W →T ′ )∈TNis
E(T ′ ).
Given a Nisnevich neighborhood (p : T ′ → T, s : W → T ′ ), we have the
isomorphism
h
p∗ : E(Tˆh ) → E(Tˆ′
).
W
s(W )
Thus, for each open subscheme j : U → W , we may evaluate E on the
ˆ
cosimplicial pro-scheme τǫX (W )(U ), giving us the presheaf of simplicial spectra
ˆ
E(τǫX (W )) on WZar :
ˆ
ˆ
E(τǫX (W ))(U ) := E(τǫX (W )(U )).
ˆ
Now suppose that E is in Spc(k). The map ˆiW : ∆∗WZar → τǫX (W )) gives us
the map of presheaves on WZar
ˆ
i∗W : E(τǫX (W )) → E(∆∗WZar ).
Similarly, the map πX,W gives the map of presheaves on WZar
ˆ
∗
πX,W
: E(∆∗X|WZar ) → E(τǫX (W )).
The main result of this section is
Documenta Mathematica 12 (2007) 71–146
80
Marc Levine
ˆ
Theorem 3.2.1. Let E be in Spc(k). Then the map i∗W : E(τǫX (W )) →
E(∆∗WZar ) is a weak equivalence for the Zariski-local model structure, i.e., for
each point w ∈ W , the map i∗W,w on the stalks at w is a weak equivalence of
the associated total space.
3.3. Proof of Theorem 3.2.1. The proof relies on two lemmas.
Lemma 3.3.1. Let i : W → X be a closed immersion in Sm/k, giving the closed
immersion A1W → A1X . For t ∈ A1 (k), we have the section it : W → A1W ,
it (w) := (w, t). Then for each E ∈ Spc(k), the maps
ˆ ∗ 1 1 ) → E(∆
ˆ ∗X,W )
i∗0 , i∗1 : E(∆
AX ,AW
are homotopic.
Proof. This is just an adaptation of the standard triangulation argument. For
each order-preserving map g = (g1 , g2 ) : [m] → [1] × [n], let
Tg : ∆m → ∆1 × ∆n ,
be the affine-linear extension of the map on the vertices
vi 7→ (vg1 (i) , vg2 (i) ).
idX × Tg induces the map
ˆ m → (∆1\
Tˆg : ∆
× ∆nX )h∆1 ×∆n
X,W
W
We note that the isomorphism (t0 , t1 ) 7→ t0 of (∆1 , v1 , v0 ) with (A1 , 0, 1) induces
an isomorphism of cosimplicial schemes
ˆ ∗1
∆
A
1
X ,AW
The maps
∼
× ∆∗X )h∆1 ×∆∗ .
= (∆1\
W
ˆ n1
Tˆg∗ : E(∆
A
1
X ,AW
ˆm )
) → E(∆
X,W
induce a simplicial homotopy T between i∗0 and i∗1 . Indeed, we have the simplicial sets ∆[n] : HomOrd (−, [n]). Let (∆1 × ∆∗ )∆[1] be the cosimplicial scheme
Y
n 7→ (∆1 × ∆n )∆[1]([n]) :=
∆1 × ∆n
s∈∆[1]([n])
where the product is ×Z . The inclusions δ0 , δ1 : [0] → [1] thus induce the maps
of cosimplicial schemes
δ0∗ , δ1∗ : (∆1 ×k ∆∗ )∆[1] → ∆1 ×k ∆∗ .
The maps Tg satisfy the identities necessary to define a map of cosimplicial
schemes
T : ∆∗ → (∆1 × ∆∗ )∆[1] .
with δ0∗ ◦ T = i0 , δ1∗ ◦ T = i1 . Applying the functor h , we see that the maps Tˆg
define the map of cosimplicial schemes
ˆ ∗X,W → (∆
ˆ ∗ 1 1 )∆[1] ,
Tˆ : ∆
AX ,AW
with δ0∗ ◦ Tˆ = ˆi0 , δ1∗ ◦ Tˆ = ˆi1 ; we then apply E.
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
81
Lemma 3.3.2. Take W ∈ Smk . Let X = AnW and let i : W → X be the
ˆ ∗ ) → E(∆∗ ) is a homotopy equivalence.
0-section. Then i∗W : E(∆
X,W
W
Proof. Let p : X → W be the projection, giving the map
ˆ ∗X,W → ∆
ˆ ∗W,W = ∆∗W
pˆ : ∆
ˆ ∗ ). Clearly ˆi∗ ◦ pˆ∗ = id, so it suffices to show that
and pˆ∗ : E(∆∗W ) → E(∆
X,W
W
pˆ∗ ◦ ˆi∗W is homotopic to the identity.
For this, we use the multiplication map µ : A1 × An → An ,
µ(t; x1 , . . . , xn ) := (tx1 , . . . , txn ).
The map µ × id∆∗ induces the map
\
µ
ˆ : (A1 ×\
AnW × ∆∗ )hA1 ×0W ×∆∗ → (AnW
× ∆∗ )h0W ×∆∗
with µ
ˆ ◦ ˆi0 = ˆiW ◦ pˆ and µ
ˆ ◦ ˆi1 = id. Since ˆi∗0 and ˆi∗1 are homotopic by
Lemma 3.3.1, the proof is complete.
To complete the proof of Theorem 3.2.1, take a point w ∈ W . Then replacing
X with a Zariski open neighborhood of w, we may assume there is a Nisnevich
neigborhood X ′ → X, s : W → X ′ of W in X such that W → X ′ is in
turn a Nisnevich neighborhood of the zero-section W → AnW , n = codimX W .
ˆ n ) is thus weakly equivalent to E(∆
ˆ nn
Since E(∆
X,W
AW ,0W ), the result follows
from Lemma 3.3.2.
Corollary 3.3.3. Suppose that E ∈ Spc(Sm/k), resp. E ∈ Spt(Sm/k) is
homotopy invariant. Then for i : W → X a closed immersion, there is a
natural isomorphism in HSpc(WZar ), resp. HSpt(WZar )
ˆ
ˆ
E(τǫX (W )) ∼
= E(τǫNi (0W ))
Here Ni is the normal bundle of the immersion i, and 0W ⊂ Ni is the 0-section.
Proof. This follows directly from Theorem 3.2.1: Since E is homotopy invariant, the canonical map
E(T ) → E(∆∗T )
is a weak equivalence for each T ∈ Sm/k. The desired isomorphism in the
respective homotopy category is constructed by composing the isomorphisms
ˆ
i∗
W
E(τǫX (W )) −−
→ E(∆∗WZar ) ← E(WZar )
i∗
0
ˆ
W
= E(0W Zar ) → E(∆∗0W Zar ) ←−
− E(τǫNi (0W )).
4. Punctured tubular neighborhoods
ˆ
Our real interest is not in the tubular neighborhood τǫX (W ), but in the puncˆ
tured tubular neighborhood τǫX (W )0 . In this section, we define this object and
discuss its basic properties.
Documenta Mathematica 12 (2007) 71–146
82
Marc Levine
4.1. Definition of the punctured neighborhood. Let i : W → X be
a closed immersion in Sm/k. We have the closed immersion of cosimplicial
pro-schemes
ˆ ∗X,W
ˆi : ∆∗W → ∆
ˆn
ˆn
giving for each n the open complement ∆
:= ∆
\ ∆n . We may pass
X\W
X,W
W
to the cofinal subcategory of Nisnevich neighborhoods of ∆nW in ∆nX ,
(p : T → ∆nX , s : ∆nW → T )
for which the diagram
T \ s(∆nW )
/T
∆nX \ ∆nW
/ ∆nX
ˆn
is cartesian, giving us the cosimplicial proscheme n 7→ ∆
X\W and the map
ˆ∗
ˆ∗
ˆj : ∆
X\W → ∆X,W ,
∗
ˆ∗
ˆ∗
which defines the “open complement” ∆
X\W of ∆W in ∆X,W . Extending this
construction to all open subschemes of X, we have the co-presheaf on WZar ,
ˆ∗
U = W \ F 7→ ∆
,
(X\F )\U
ˆ
τǫX (W )0 .
which we denote by
Let ∆n(X\W )Zar be the constant co-presheaf on WZar with value ∆nX\W , giving
the cosimplicial co-presheaf ∆∗(X\W )Zar . The maps
ˆ∗
ˆ∗
ˆjU : ∆
(X\F )\U → ∆(X\F ),U
ˆ
ˆ
∗
ˆ∗
define the map ˆj : τǫX (W )0 → τǫX (W ). The maps ∆
U \W ∩U → ∆X\W give us
the map
ˆ
π : τǫX (W )0 → ∆∗X\W
where we view ∆∗X\W as the constant co-sheaf on WZar .
ˆ
To give a really useful result on the presheaf E(τǫX (W )0 ), we will need to
impose additional conditions on E. These are
(1) E is homotopy invariant
(2) E satisfies Nisnevich. excision
One important consequence of these properties is the purity theorem of MorelVoevodsky:
Theorem 4.1.1 (Purity [34, theorem 2.23]). Suppose E ∈ Spt(k) is homotopy
invariant and satisfies Nisnevich excision. Let i : W → X be a closed immersion in Sm/k and s : W → Ni the 0-section of the normal bundle. Then there
is an isomorphism in HSpt(WZar )
E WZar (X) → E WZar (Ni )
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
83
Let E(X|WZar ) be the presheaf on WZar
W \ F 7→ E(X \ F )
and E(X \ W ) the constant presheaf.
Let
ˆ
res : E(X|WZar ) → E(τǫX (W ))
ˆ
res0 : E(X \ W ) → E(τǫX (W )0 )
ˆ
ˆ
be the pull-back by the natural maps τǫX (W )(W \ F ) → X \ F , τǫX (W )0 →
∗
ˆ
X \W . Let E ∆W (τǫX (W )) ∈ Spt(WZar ) be the homotopy fiber of the restriction
map
ˆj : E(τǫXˆ (W )) → E(τǫXˆ (W )0 ).
The commutative diagram in Spt(WZar )
E(X|WZar )
j∗
/ E(X \ W )
res
ˆ
res0
E(τǫX (W ))
ˆ
j∗
/ E(τ Xˆ (W )0 )
ǫ
induces the map of homotopy fiber sequences
E WZar (X)
/ E(X \ W )
res
ψ
∗
j∗
/ E(X|WZar )
res0
ˆ
E ∆W (τǫX (W ))
/ E(τ Xˆ (W ))
ǫ
j∗
/ E(τ Xˆ (W )0 )
ǫ
ˆ
We can now state the main result for E(τǫX (W )0 ).
Theorem 4.1.2. Suppose that E ∈ Spt(k) is homotopy invariant and satisfies
Nisnevich excision. Let i : W → X be a closed immersion in Sm/k. Then the
map ψ is a Zariski local weak equivalence.
Proof. Let i∆∗ : ∆∗W → ∆∗X be the immersion id × i. For U = W \ F ⊂ W ,
ˆ
τǫX (W )0 (U ) is the cosimplicial pro-scheme with n-cosimplices
ˆ
n
ˆn
τǫX (W )0 (U )n = ∆
X\F,U \ ∆U
so by Nisnevich excision we have the natural isomorphism
∗
∗
ˆ
α : E ∆WZar (∆∗X|WZar ) → E ∆W (τǫX (W )),
∗
where E ∆WZar (∆∗X|WZar )(W \F ) is the total spectrum of the simplicial spectrum
n
n 7→ E ∆W \F (∆nX\F ).
The homotopy invariance of E implies that the pull-back
n
E W \F (X \ F ) → E ∆W \F (∆nX\F )
Documenta Mathematica 12 (2007) 71–146
84
Marc Levine
is a weak equivalence for all n, so we have the weak equivalence
∗
β : E WZar (X) → E ∆WZar (∆∗X|WZar ).
It follows from the construction that ψ = αβ, completing the proof.
Corollary 4.1.3. There is a distinguished triangle in HSpt(WZar )
ˆ
E WZar (X) → E(WZar ) → E(τǫX (W )0 )
ˆ
Proof. By Theorem 3.2.1, the map ˆi∗ : E(τǫX (W )) → E(∆∗WZar ) is a weak
equivalence; using homotopy invariance again, the map
E(WZar ) → E(∆∗WZar )
is a weak equivalence. Combining this with Theorem 4.1.2 yields the result.
For homotopy invariant E ∈ Spt(k), we let
ˆ
ˆ
φE : E(τǫNi (0W )) → E(τǫX (W )).
be the isomorphism in HSpt(WZar ) given by corollary 3.3.3.
Corollary 4.1.4. Suppose that E ∈ Spt(k) is homotopy invariant and satisfies Nisnevich excision. Let i : W → X be a closed immersion in Sm/k and
let Ni0 = Ni \ 0W .
(1) The restriction maps
ˆ
res : E(Ni /WZar ) → E(τǫNi (0W ))
ˆ
res0 : E(Ni0 /WZar ) → E(τǫNi (0W )0 )
are weak equivalences in Spt(WZar ).
(2) There is a canonical isomorphism in HSpt(WZar )
ˆ
ˆ
φ0E : E(τǫNi (0W )0 ) → E(τǫX (W )0 )
(3) Consider the diagram (in HSpt(WZar )):
E 0W Zar (Ni )
/ E(N 0 /WZar )
/ E(Ni /WZar )
i
res0E
resE
E 0W Zar (Ni )
/ E(τ Nˆi (0 ))
W
ǫ
π
E WZar (X)
jˆ
N
∗
φ0E
φE
/ E(τ Xˆ (W ))
ǫ O
ˆ
j∗
/ E(X|WZar )
/ E(τ Xˆ (W )0 )
ǫ O
res0E
resE
E WZar (X)
/ E(τ Nˆi (0 )0 )
W
ǫ
j∗
/ E(X \ W )
The first and last rows are the homotopy fiber sequences defining the presheaves
E 0W Zar (Ni ) and E WZar (X), respectively, the second row and third rows are the
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
85
distinguished triangles of Theorem 4.1.2, and π is the Morel-Voevodsky purity
isomorphism. Then this diagram commutes and each triple of vertical maps
defines a map of distinguished triangles.
Proof. It follows directly from the weak equivalence (in Theorem 4.1.2) of the
homotopy fiber of
ˆj ∗ : E(τǫXˆ (W )) → E(τǫXˆ (W )0 )
with E WZar (X) that the triple (id, resE , res0E ) defines a map of distinguished
triangles. The same holds for the map of the first row to the second row; we
now verify that this latter map is an isomorphism of distinguished triangles.
For this, let s : W → Ni be the zero-section. We have the isomorphism
ˆ
i∗W : E(τǫNi (0W )) → E(WZar ) defined as the diagram of weak equivalences
i∗
∆∗
ˆ
ι
W
0∗
E(τǫNi (0W )) −−−−Zar
−→ E(∆∗WZar ) ←−
− E(WZar ).
From this, it is easy to check that the diagram
resE
/ E(τ Nˆi (0 ))
E(Ni /WZar )
W
ǫ
OOO
OOO
OOO
i∗
W
OOO
s∗
'
E(WZar )
commutes in HSpt(WZar ). As E is homotopy invariant, s∗ is an isomorphism,
hence resE is an isomorphism as well. This completes the proof of (1).
The proof of (2) and (3) uses the standard deformation diagram. Let µ
¯ : Y¯ →
X × A1 be the blow-up of X × A1 along W , let µ
¯−1 [X × 0] denote the proper
transform, and let µ : Y → X × A1 be the open subscheme Y¯ \ µ
¯−1 [X × 0].
Let p : Y → A1 be p2 ◦ µ. Then p−1 (0) = Ni , p−1 (1) = X × 1 = X, and Y
contains the proper transform µ
¯−1 [W × A1 ], which is mapped isomorphically
1
1
by µ to W × A ⊂ X × A . We let ˜i : W × A1 → Y be the resulting closed
immersion. The restriction of ˜i to W × 0 is the zero-section s : W → Ni and
the restriction of ˜i to W × 1 is i : W → X. The resulting diagram is
(4.1.1)
W
i0
i1
/ W × A1 o
s
W
˜i
Ni
i0
/Y o
i
i1
X
p
p0
0
i0
/ A1 o
p1
i1
1
Documenta Mathematica 12 (2007) 71–146
86
Marc Levine
Together with Theorem 4.1.2, diagram (4.1.1) gives us two maps of distinguished triangles:
h
i
ˆ
ˆ
E 0W ×A1 Zar (Y ) → E(τǫY (W × A1 )) → E(τǫY (W × A1 )0 )
i∗
and
h
h
1
−→
i
ˆ
ˆ
E WZar (X) → E(τǫX (W )) → E(τǫX (W )0 )
i
ˆ
ˆ
E 0W ×A1 Zar (Y ) → E(τǫY (W × A1 )) → E(τǫY (W × A1 )0 )
i∗
0
−→
i
h
ˆ
ˆ
E 0W Zar (Ni ) → E(τǫNi (0W )) → E(τǫNi (0W )0 )
As above, we have the commutative diagram
ˆ
E(τǫY (W × A1 ))
i∗
0
i∗
W ×A1
/ E(τ Nˆi (0 ))
W
ǫ
i∗
W
E(W × A1 )
i∗
0
/ E(W ).
As E is homotopy invariant, the maps i∗W , i∗W ×A1 and i∗0 : E(W × A1 ) → E(W )
are isomorphisms, hence
ˆ
ˆ
i∗0 : E(τǫY (W × A1 )) → E(τǫNi (0W ))
is an isomorphism. Similarly,
ˆ
ˆ
i∗1 : E(τǫY (W × A1 )) → E(τǫX (W ))
is an isomorphism. The proof of the Morel-Voevodsky purity theorem [34,
Theorem 2.23] shows that
i∗0 : E 0W ×A1 Zar (Y ) → E 0W Zar (Ni )
i∗1 : E 0W ×A1 Zar (Y ) → E WZar (X)
are weak equivalences; the purity isomorphism π is by definition i∗1 ◦ (i∗0 )−1 .
Thus, both i∗0 and i∗1 define isomorphisms of distinguished triangles, and
ˆ
ˆ
i∗1 ◦ (i∗0 )−1 : E(τǫNi (0W )) → E(τǫX (W ))
is the map φE . Defining φ0E to be the isomorphism
ˆ
ˆ
i∗1 ◦ (i∗0 )−1 : E(τǫNi (0W )0 ) → E(τǫX (W )0 )
proves both (2) and (3).
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
87
Remarks 4.1.5. (1) It follows from the construction of φE and φ0E that both of
these maps are natural in E.
(2) The maps φ0E are natural in the closed immersion i : W → X in the
following sense: Suppose we have closed immersions ij : Wj → Xj , j = 1, 2
and a morphism f : (W1 , X1 ) → (W2 , X2 ) of pairs of immersions such that f
restricts to a morphism X1 \ W1 → X2 \ W2 . Fix E and let φ0jE be the map
corresponding to the immersions ij We have the evident maps
ˆ
ˆi
N
1
ˆ
ι : τǫX1 (W1 )0 → τǫX2 (W2 )0
η : τǫ
ˆi
N
2
(W1 )0 → τǫ
(W2 )0
Then the diagram
ˆi
N
2
E(τǫ
(W )0 )
φ2E
/ E(τ Yˆ (W )0 )
ǫ
η∗
ι∗
ˆi
N
1
E(τǫ
(W )0 )
φ1E
/ E(τ Xˆ (W )0 )
ǫ
commutes. Indeed, the map f induces a map of deformation diagrams.
5. The exponential map
′
If i : M → M is a submanifold of a differentiable manifold M , there is a
diffeomorphism exp of the normal bundle NM ′ /M of M ′ in M with the tubular
neighborhood τǫM (M ′ ). In addition, exp restricts to a diffeomorphism exp0 of
the punctured normal bundle NM ′ /M \ 0M ′ with the punctured tubular neighborhood τǫM (M ′ ) \ M ′ . Classically, this has been used to define the boundary map in the Gysin sequence for M ′ → M , by using the restriction map
exp0∗ : H ∗ (M \ M ′ ) → H ∗ (NM ′ /M \ 0M ′ ) followed by the Thom isomorphism
H ∗ (NM ′ /M \ 0M ′ ) ∼
= H ∗−d (M ′ ).
In this section, we use our punctured tubular neighborhood to construct a
purely algebraic version of the classical exponential map, at least for the associated suspension spectra. We will use this in section 11 to define a purely
algebraic version of the gluing of Riemann surfaces along boundary components.
5.1. Let i : W → X be a closed immersion in Sm/k with normal bundle
p : Ni → W . We have the map
exp : Ni → X
in Spc(k), defined as the composition Ni → W → X. We also have the
Morel-Voevodsky purity isomorphism
π : Th(Ni ) → X/(X \ W )
in H(k). In fact:
Documenta Mathematica 12 (2007) 71–146
88
Marc Levine
Lemma 5.1.1. The diagram
(5.1.1)
q′
Ni
/ Th(Ni )
exp
π
X
/ X/(X \ W )
q
commutes in H(k).
Proof. As we have already seen, the construction of the purity isomorphism π
relies on the deformation to the normal bundle; we retain the notation from the
proof of corollary 4.1.4. We have the total space Y → A1 of the deformation.
The fiber Y0 over 0 ∈ A1 is canonically isomorphic to Ni and the fiber Y1 over
1 is canonically isomorphic to X; the inclusions W × 0 → Y0 , W × 1 → Y1
are isomorphic to the zero-section s : W → Ni and the original inclusion
i : W → X, respectively. The proper transform µ−1 [W × A1 ] is isomorphic
to W × A1 , giving the closed immersion ι : W × A1 → Y . The diagram thus
induces maps in Spc(k):
¯i0 : Th(Ni ) → Y /(Y \ W × A1 )
¯i1 : X/X \ W → Y /(Y \ W × A1 )
which are isomorphisms in H(k) (see [34, Thm. 2.23]); the purity isomorphism
¯
is by definition π := ¯i−1
1 ◦ i0 .
We have the commutative diagram in Spc(k):
id
i0
W
O
p
/ W × A1 o
s
p1
/
W
i1
ι
i
/Y o
X
/ Y /(Y \ W × A1 ) o
X/X \ W
Ni
q′
q
Th(Ni )
¯i0
¯i1
from which the result follows directly.
Remark 5.1.2. Since we have the homotopy cofiber sequences:
Ni \ 0W → Ni → Th(Ni ) → Σ(Ni \ 0W )+
X \ W → X → X/(X \ W ) → Σ(X \ W )+
the diagram (5.1.1) induces a map
Σ(Ni \ 0W )+ → Σ(X \ W )+
in H(k), however, this map is not uniquely determined, hence is not canonical.
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
89
5.2. The construction. In this section we define a canonical map
exp0 : Σ∞ (Ni \ 0W )+ → Σ∞ (X \ W )+
in SHA1 (k) which yields the map of distinguished triangles in SHA1 (k):
/ Σ∞ Ni+
Σ∞ (Ni \ 0W )+
/ Σ∞ Th(Ni )
exp
exp0
π
/ Σ∞ X/(X \ W )
/ Σ∞ X+
Σ∞ (X \ W )+
To define exp0 , we apply Corollary 4.1.4 with E a fibrant model of Σ∞ (X \W )+ .
Denote the composition
res0
ˆ
E
E(X \ W ) −−−→
E(τǫX (W )0 )(W )
(φ0 )−1
(res0 )−1
ˆ
E
−−−
−−→ E(τǫNi (0W )0 )(W ) −−−E−−→ E(Ni0 )
by exp0∗
E . Since E is fibrant, we have canonical isomorphisms
0
, E)
π0 E(Ni0 ) ∼
= HomSHA1 (k) (Σ∞ Ni+
∞ 0
∼
= HomSHA1 (k) (Σ Ni+ , Σ∞ (X \ W )+ )
π0 E(X \ W ) ∼
= HomSHA1 (k) (Σ∞ (X \ W )+ , E)
∼ HomSH (k) (Σ∞ (X \ W )+ , Σ∞ (X \ W )+ )
=
A1
so exp0∗
E induces the map
HomSHA1 (k) (Σ∞ (X \ W )+ , Σ∞ (X \ W )+ )
exp0∗
0
, Σ∞ (X \ W )+ ).
−−−E→ HomSHA1 (k) (Σ∞ Ni+
We set
exp0 := exp0∗
E (id).
To finish the construction, we show
Proposition 5.2.1. The diagram, with rows the evident homotopy cofiber sequences,
Σ∞ (Ni \ 0W )+
/ Σ∞ Ni+
exp
exp0
Σ∞ (X \ W )+
/ Σ∞ Th(Ni )
∂
/ ΣΣ∞ (Ni \ 0W )+
Σ exp0
π
/ Σ∞ X+
/ Σ∞ X/(X \ W )
∂
/ ΣΣ∞ (X \ W )+
commutes in SHA1 (k).
Documenta Mathematica 12 (2007) 71–146
90
Marc Levine
Proof. It suffices to show that, for all fibrant E ∈ Spt(k), the diagram formed
by applying HomSHA1 (k) (−, E) to our diagram commutes. This latter diagram
is the same as applying π0 to the diagram
(5.2.1)
E(Ni \ 0W ) o
O
exp0∗
E(X \ W ) o
E 0W (Ni ) o
O
E(Ni ) o
O
exp∗
∂
ΩE(Ni \ 0W ))
O
Ω exp0∗
π∗
E W (X) o
E(X) o
ΩE(X \ W )
∂
where the rows are the evident homotopy fiber sequences. It follows by the
definition of exp0 and exp that this diagram is just the “outside” of the diagram
in Corollary 4.1.4(3), extended to make the distinguished triangles explicit.
Thus the diagram (5.2.1) commutes, which finishes the proof.
Remark 5.2.2. The exponential maps exp and exp0 are natural with respect
i′
i
to maps of closed immersions f : (W ′ −
→ X ′ ) → (W −
→ X) satisfying
cartesian condition of remark 4.1.5(2). This follows from the naturality of
isomorphisms φE , φ0E described in Remark 4.1.5, and the functoriality of
(punctured) tubular neighborhood construction.
the
the
the
6. Neighborhoods of normal crossing schemes
We extend our results to the case of a strict normal crossing divisor W ⊂ X
by using a Mayer-Vietoris construction.
6.1. Normal crossing schemes. Let D be a reduced effective Cartier divisor on a smooth k-scheme X with irreducible components D1 ,. . ., Dm . For
each I ⊂ {1, . . . , m}, we set
DI := ∩i∈I Di
We let i : D → X the inclusion. For each I 6= ∅, we let ιI : DI → D,
iI : DI → X be the inclusions; for I ⊂ J ⊂ {1, . . . , m} we have as well the
inclusion ιI,J : DJ → DI .
Recall that D is a strict normal crossing divisor if for each I, DI is smooth
over k and codimX DI = |I|.
We extend this notion a bit: We call a closed subscheme D ⊂ X a strict normal
crossing subscheme if X is in Sm/k and, locally on X, there is a smooth locally
closed subscheme Y ⊂ X containing D such that D is a strict normal crossing
divisor on Y
6.2. The tubular neighborhood. Let D ⊂ X be a strict normal crossing
subscheme with irreducible components D1 , . . . , Dm . For each I ⊂ {1, . . . , m},
ˆ
I 6= ∅, we have the tubular neighborhood co-presheaf τǫX (DI ) on DI . The
various inclusions ιI,J give us the maps of co-presheaves
ˆ
ˆ
ˆιI,J : ιI,J∗ (τǫX (DJ )) → τǫX (DI );
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
91
pushing forward by the maps ιI yields the diagram of co-presheaves on DZar
(with values in cosimplicial pro-objects of Sm/k)
ˆ
I 7→ ιI∗ (τǫX (DI ))
(6.2.1)
indexed by the non-empty I ⊂ {1, . . . , m}. We have as well the diagram of
identity co-presheaves
I 7→ ιI∗ (DIZar )
(6.2.2)
as well as the diagram
I 7→ ιI∗ (∆∗DIZar )
(6.2.3)
ˆ
We denote these diagrams by τǫX (D), D• and ∆∗D• , respectively. The projecˆ
tions pI : ∆∗DIZar → DIZar and the closed immersions ˆιDI : ∆∗DIZar → τǫX (DI )
yield the natural transformations
ˆ
ιD
p•
ˆ
•
D• ←− ∆∗D• −−→
τǫX (D).
Now take E ∈ Spt(k). Applying E to the diagram (6.2.1) yields the diagram
of presheaves on DZar
ˆ
I 7→ ιI∗ (E(τǫX (DI )))
Similarly, applying E to (6.2.2) and (6.2.3) yields the diagrams of presheaves
on DZar
I 7→ ιI∗ (E(DIZar ))
and
I 7→ ιI∗ (E(∆∗DIZar )).
Definition 6.2.1. For D ⊂ X a strict normal crossing subscheme and E ∈
Spt(k), set
ˆ
ˆ
E(τǫX (D)) := holim ιI∗ (E(τǫX (DI ))).
I6=∅
Similarly, set
E(D• ) := holim ιI∗ (E(DI ))
I6=∅
E(∆∗D• )
:= holim ιI∗ (E(∆∗DI ))
I6=∅
The natural transformations ˆιD and p• yield the maps of presheaves on DZar
p∗
ˆ
ι∗
ˆ
•
D
E(D• ) −→
E(∆∗D• ) ←−
− E(τǫX (D)).
Proposition 6.2.2. Suppose E ∈ Spt(Sm/k) is homotopy invariant and satisfies Nisnevich excision. Then the maps ˆι∗D and p∗• are Zariski-local weak
equivalences.
Documenta Mathematica 12 (2007) 71–146
92
Marc Levine
Proof. The maps p∗I are pointwise weak equivalences by homotopy invariance.
By Theorem 3.2.1, the maps ˆιDI are Zariski-local weak equivalences. Since
the homotopy limits are finite, the stalk of each homotopy limit is weakly
equivalence to the homotopy limit of the stalks. By [8] this suffices to conclude
that the map on the homotopy limits is a Zariski-local weak equivalence.
ˆ
Remark 6.2.3. One could also attempt a more direct definition of τǫX (D) by
just using our definition in the smooth case i : W → X and replacing the
smooth W with the normal crossing scheme D, in other words, the co-presheaf
on DZar
ˆ∗
D \ F 7→ ∆
X\F,D\F .
ˆ
ˆ
Labeling this choice τǫX (D)naive , and considering τǫX (D)naive as a constant diagram, we have the evident map of diagrams
ˆ
ˆ
φ : τǫX (D) → τǫX (D)naive
We were unable to determine if φ induces a weak equivalence after evaulation
on E ∈ Spt(k), even assuming that E is homotopy invariant and satisfies
Nisnevich excision. We were also unable to construct such an E for which φ
fails to be a weak equivalence.
6.3. The punctured tubular neighborhood. To define the punctured
ˆ
tubular neighborhood τǫX (D)0 , we proceed as follows: Fix a subset I ⊂
{1, . . . , m}, I 6= ∅. Let p : X ′ → X, s : DI → X ′ be a Nisnevich neighborhood
of DI in X, and let DX ′ = p−1 (D). Sending X ′ → X to ∆nDX ′ gives us the
ˆn
ˆn
ˆn
pro-scheme ∆
D⊂X,DI , and the closed immersion ∆D⊂X,DI → ∆X,DI . Varying
∗
ˆ
n, we have the cosimplicial pro-scheme ∆
D⊂X,DI , and the closed immersion
∗
∗
ˆ
ˆ
.
→∆
∆
D⊂X,DI
X,DI
Take a closed subset F ⊂ DI , and let U := DI \ F . As in the definition of the
punctured tubular neighborhood of a smooth closed subscheme in section 4.1,
we pass to the appropriate cofinal subcategory of Nisnevich neighborhoods to
ˆn
ˆn
show that the open complements ∆
\∆
for varying n form a
X\F,U
D\F ⊂X\F,U
cosimplicial pro-scheme
ˆn
ˆn
n 7→ ∆
X\F,U \ ∆D\F ⊂X\F,U .
Similarly, we set
ˆ
ˆ∗
ˆ∗
τǫX (D, DI )0 (U ) := ∆
X\F,U \ ∆D\F ⊂X\F,U .
ˆ
This forms the co-presheaf τǫX (D, DI )0 on DIZar . The open immersions
ˆn
ˆn
ˆn
ˆjI (U )n : ∆
→∆
\∆
X\F,U
D\F ⊂X\F,U
X\F,U
define the map
ˆjI (U ) : τǫXˆ (D, DI )0 (U ) → τǫXˆ (DI )(U ),
giving the map of co-presheaves
ˆjI : τǫXˆ (D, DI )0 → τǫXˆ (DI ).
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
93
ˆ∗
ˆ∗
For J ⊂ I, we have the map ˆιJ,I : ∆
X,DI → ∆X,DJ and
ˆ∗
ˆ∗
ˆι−1
J,I (∆D⊂X,DJ ) = ∆D⊂X,DI .
ˆ
ˆ
Thus we have the map ˆι0J,I : τǫX (D, DI )0 → τǫX (D, DJ )0 and the diagram of
co-presheaves on DZar
ˆ
I 7→ ιI∗ (τǫX (D, DI )0 )
(6.3.1)
ˆ
which we denote by τǫX (D)0 . The maps ˆjI define the map
ˆj : τǫXˆ (D)0 → τǫXˆ (D).
ˆ
The projection maps πI : τǫX (DI ) → X (where we consider X as the constant
ˆ
co-presheaf on DIZar ) restrict to maps πI0 : τǫX (D, DI )0 → X \ D, which in turn
induce the map
ˆ
π 0 : τǫX (D)0 → X \ D,
where we consider X \ D the constant diagram of constant co-presheaves on
DZar .
ˆ
Definition 6.3.1. For E ∈ Spt(k), let E(τǫX (D)0 ) be the presheaf on DZar ,
ˆ
E(τǫX (D)0 ) :=
holim
ˆ
∅6=I⊂{1,...,m}
ιI∗ E(τǫX (D, DI )0 ).
The map ˆj defines the map of presheaves
ˆj ∗ : E(τǫXˆ (D)) → E(τǫXˆ (D)0 ).
ˆ
We let E DZar (τǫX (D)) denote the homotopy fiber of ˆj ∗ . Via the commutative
diagram
E(X \ F )
j∗
/ E(X \ D)
π∗
ˆ
E(τǫX (D))(D \ F )
π 0∗
ˆ
j∗
/ E(τ Xˆ (D))0 (D \ F )
ǫ
we have the canonical map
ˆ
∗
πD
: E DZar (X) → E DZar (τǫX (D)).
∗
We want to show that the map πD
is a weak equivalence, assuming that E is
homotopy invariant and satisfies Nisnevich excision. We first consider a simpler
situation. We begin by noting the following
Lemma 6.3.2. Let n0 denote the category of non-empty subsets of {1, . . . , n}
with maps the inclusions, let C be a small category and let F : C × n0 →
op
op
SptOrd be a functor. Let holimn0 F : C → SptOrd be the functor with
Documenta Mathematica 12 (2007) 71–146
94
Marc Levine
value the simplicial spectrum m 7→ holimn0 F (i × [m]) at i ∈ C. There is a
isomorphism
Tot(holim
F ) → holim
Tot(F ).
n
n
0
0
op
in HSpt(C ).
Proof. Letting n be the category of all subsets of {1, . . . , n} (including the
empty set), we may extend F to F ∗ : n → Spt(Ordop ) by F ∗ (∅) = 0.
Similarly, given a functor G : n → Spt, we may extend G to G♮ : n+1
→ Spt
0
by G♮ (I) = 0, G♮ (I ∪ {n + 1}) = G(I) for I ⊂ {1, . . . , n}. We define the iterated
homotopy fiber of G, fibn G ∈ Spt, by
G♮ .
hofibn (G) := holim
n+1
0
One easily checks that for a map g : A → B of spectra,
Documenta Math.
Motivic Tubular Neighborhoods
Marc Levine
Received: November 12, 2005
Revised: January 2, 2007
Communicated by Alexander Merkurjev
Abstract. We construct motivic versions of the classical tubular
neighborhood and the punctured tubular neighborhood, and give applications to the construction of tangential base-points for mixed Tate
motives, algebraic gluing of curves with boundary components, and
limit motives.
2000 Mathematics Subject Classification: Primary 14C25; Secondary
55P42, 18F20, 14F42
Keywords and Phrases: Limit motives, moduli of curves, Tate motives
1. Introduction
Let i : A → B be a closed embedding of finite CW complexes. One useful
fact is that A admits a cofinal system of neighborhoods T in B with A → T a
deformation retract. This is often used in the case where B is a differentiable
manifold, showing for example that A has the homotopy type of the differentiable manifold T . This situation occurs in algebraic geometry, for instance
in the case of the inclusion of the special fiber in a degeneration of smooth
varieties X → C over the complex numbers.
To some extent, one has been able to mimic this construction in purely algebraic
terms. The rigidity theorems of Gillet-Thomason [14], extended by Gabber
(details appearing a paper of Fujiwara [13]) indicated that, at least through
the eyes of torsion ´etale sheaves, the topological tubular neighborhood can be
replaced by the Hensel neighborhood. However, basic examples of non-torsion
phenomena, even in the ´etale topology, show that the Hensel neighborhood
cannot always be thought of as a tubular neighborhood, perhaps the simplest
example being the sheaf Gm .1
Our object in this paper is to construct an algebraic version of the tubular
neighborhood which has the basic properties of the topological construction,
at least for a reasonably large class of cohomology theories. It turns out that
1If O is a local ring with residue field k and maximal ideal m, the surjection G (O) →
m
Gm (k) has kernel (1 + m)× , which is in general non-zero, even for O Hensel
Documenta Mathematica 12 (2007) 71–146
72
Marc Levine
a “homotopy invariant” version of the Hensel neighborhood does the job, at
least for theories which are themselves homotopy invariant. If one requires in
addition that the given cohomology theory has a Mayer-Vietoris property for
the Nisnevich topology, then one also has an algebraic version of the punctured
tubular neighborhood. We extend these constructions to the case of a (reduced)
strict normal crossing subscheme by a Mayer-Vietoris procedure, giving us
the tubular neighborhood and punctured tubular neighborhood of a normal
crossing subscheme of a smooth k-scheme.
Morel and Voevodsky have constructed an algebro-geometric version of homotopy theory, in the setting of presheaves of spaces or spectra on the category
of smooth varieties over a reasonable base scheme B; we concentrate on the
A1 -homotopy category of spectra, SHA1 (B). For a map f : X → Y , they
construct a pair of adjoint functors
Rf∗ : SHA1 (X) → SHA1 (Y )
Lf ∗ : SHA1 (Y ) → SHA1 (X).
If we have a closed immersion i : W → X with open complement j : U → X,
then one has the functor
Li∗ Rj∗ : SHA1 (U ) → SHA1 (W )
One of our main results is that, in case W is a strict normal crossing subscheme
of a smooth X, the restriction of Li∗ Rj∗ E to a Zariski presheaf on W can be
viewed as the evaluation of E on the punctured tubular neighborhood of W in
X.
Consider a morphism p : X → A1 and take i : W → X to be the inclusion
of p−1 (0). Following earlier constructions of Spitzweck [43], Ayoub has constructed a “unipotent specialization functor” in the motivic setting, essentially
(in the case of a semi-stable degeneration) by evaluating Li∗ Rj∗ E on a cosimplicial version of the appropriate path space on Gm with base-point 1. Applying
the same idea to our tubular neighborhood construction gives a model for this
specialization functor, again only as a Zariski presheaf on p−1 (0).
Ayoub has also defined a motivic monodromy operator and monodromy sequence involving the unipotent specialization functor and the functor Li∗ Rj∗ ,
for theories with Q-coefficients that satisfy a certain additional condition (see
definition 9.2.2). We give a model for this construction by combining our
punctured tubular neighborhood with a Q-linear version of the Gm -path space
mentioned above. We conclude with an application of our constructions to the
moduli spaces of smooth curves and a construction of a specialization functor for category of mixed Tate motives, which in some cases yields a purely
algebraic construction of tangential base-points. Of course, the construction
of Ayoub, when restricted to the triangulated category of Tate motives, also
gives such a specialization functor, but we hope that the explicit nature of our
construction will be useful for applications.
We have left to another paper the task of checking the compatibilities of our
constructions with others via the appropriate realization functor. As we have
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
73
already mentioned, our punctured tubular neighborhood construction is comparable with the motivic version of the functor Li∗ Rj∗ for the situation of a
normal crossing scheme i : D → X with open complement j : X \ D → X;
this should imply that it is a model for the analogous functor after realization.
Similarly, our limit cohomology construction should transform after realization
to the appropriate version of the sheaf of vanishing cycles, at least in the case of
a semi-stable degeneration, and should be comparable with the constructions
of Rappaport-Zink [37] as well as the limit mixed Hodge structure of Katz [22]
and Steenbrink [44]. Our specialization functor for Tate motives should be
compatible with the Betti, ´etale and Hodge realizations; similarly, realization
functors applied to our limit motive should yield for example the limit mixed
Hodge structure. We hope that our rather explicit construction of the limiting
motive will be useful in giving a geometric view to the limit mixed Hodge structure of a semi-stable degeneration but we have not attempted an investigation
of these issues in this paper.
My interest in this topic began as a result of several discussions on limit motives with Spencer Bloch and H´el`ene Esnault, whom I would like to thank for
their encouragement and advice. I would also like to thank H´el`ene Esnault for
clarifying the role of the weight filtration leading to the exactness of ClemensSchmidt monodromy sequence (see Remark 9.3.6). An earlier version of our
constructions used an analytic (i.e. formal power series) neighborhood instead
of the Hensel version now employed; I am grateful to Fabien Morel for suggesting this improvement. Finally, I want to thank Joseph Ayoub for explaining
his construction of the nearby cycles functor; his comments suggested to us
the use of the cosimplicial path space in our construction of limit cohomology.
In addition, Ayoub noticed a serious error in our first attempt at constructing the monodromy sequence; the method used in this version is following his
suggestions and comments. Finally, we would like to thank the referee for giving unusually thorough and detailed comments and suggestions, which have
substantially improved this paper. In particular, the material in sections 7
comparing our construction with the categorical ones of Morel-Voevodsky, as
well as the comparison with Ayoub’s specialization functor and monodromy
sequence in section 8.3 and section 9 was added following the suggestion of the
referee, who also supplied the main ideas for the proofs.
2. Model structures and other preliminaries
2.1. Presheaves of simplicial sets. We recall some facts on the model
structures in categories of simplicial sets, spectra, associated presheaf categories
and certain localizations. For details, we refer the reader to [17] and[19].
For a small category I and category C, we will denote the category of functors
from I to C by C I .
We let Ord denote the category with objects the finite ordered sets [n] :=
{0, . . . , n} (with the standard ordering) and morphisms the order-preserving
op
maps of sets. For a category C, the functor categories C Ord , C Ord are the
Documenta Mathematica 12 (2007) 71–146
74
Marc Levine
categories of cosimplicial objects of C, resp. simplicial objects of C. For C =
op
Sets, we have the category of simplicial sets, Spc := SetsOrd , and similarly
for C the category of pointed sets, Sets∗ , the category of pointed simplicial sets
op
.
Spc∗ := SetsOrd
∗
We give Spc and Spc∗ the standard model structures: cofibrations are
(pointed) monomorphisms, weak equivalences are weak homotopy equivalences
on the geometric realization, and fibrations are detemined by the right lifting property (RLP) with respect to trivial cofibrations; the fibrations are then
exactly the Kan fibrations. We let |A| denote the geometric realization, and
[A, B] the homotopy classes of (pointed) maps |A| → |B|.
For an essentially small category C, we let Spc(C) be the category of presheaves
of simplicial sets on C. We give Spc(C) the so-called injective model structure,
that is, the cofibrations and weak equivalences are the pointwise ones, and the
fibrations are determined by the RLP with respect to trivial cofibrations. We
let HSpc(C) denote the associated homotopy category (see [17] for details on
these model structures for Spc and Spc(C)).
2.2. Presheaves of spectra. Let Spt denote the category of spectra. To
fix ideas, a spectrum will be a sequence of pointed simplicial sets E0 , E1 , . . .
together with maps of pointed simplicial sets ǫn : S 1 ∧ En → En+1 . Maps of
spectra are maps of the underlying simplicial sets which are compatible with
the attaching maps ǫn . The stable homotopy groups πns (E) are defined by
πns (E) := lim [S m+n , Em ].
m→∞
The category Spt has the following model structure: Cofibrations are maps
f : E → F such that E0 → F0 is a cofibration, and for each n ≥ 0, the map
a
S 1 ∧ Fn → Fn+1
En+1
S 1 ∧En
is a cofibration. Weak equivalences are the stable weak equivalences, i.e., maps
f : E → F which induce an isomorphism on πns for all n. Fibrations are
characterized by having the RLP with respect to trivial cofibrations. We write
SH for the homotopy category of Spt.
For X ∈ Spc∗ , we have the suspension spectrum Σ∞ X := (X, ΣX, Σ2 X, . . .)
with the identity bonding maps. Dually, for a spectrum E := (E0 , E1 , . . .) we
have the 0-space Ω∞ E := limn Ωn En . These operations form a Quillen pair
of adjoint functors (Σ∞ , Ω∞ ) between Spc∗ and Spt, and thus induce adjoint
functors on the homotopy categories.
Let C be a category. A functor E : C op → Spt is called a presheaf of spectra on
C.
We use the following model structure on the category of presheaves of spectra (see [19]): Cofibrations and weak equivalences are given pointwise, and
fibrations are characterized by having the RLP with respect to trivial cofibrations. We denote this model category by Spt(C), and the associated homotopy
category by HSpt(C).
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
75
As a particular example, we have the model category of simplicial spectra
op
SptOrd = Spt(Ord). We have the total spectrum functor
Tot : Spt(Ord) → Spt
which preserves weak equivalences. The adjoint pair (Σ∞ , Ω∞ ) extend pointwise to define a Quillen pair on the presheaf categories and an adjoint pair on
the homotopy categories.
Let B be a noetherian separated scheme of finite Krull dimension. We let
Sm/B denote the category of smooth B-schemes of finite type over B. We
often write Spc(B) and HSpc(B) for Spc(Sm/B) and HSpc(Sm/B), and
write Spt(B) and HSpt(B) for Spt(Sm/B) and HSpt(Sm/B).
For Y ∈ Sm/B, a subscheme U ⊂ Y of the form Y \ ∪α Fα , with {Fα } a
possibly infinite set of closed subsets of Y , is called essentially smooth over B;
the category of essentially smooth B-schemes is denoted Smess .
2.3. Local model structure. If the category C has a topology, there is often
another model structure on Spc(C) or Spt(C) which takes this into account. We
consider the case of the small Nisnevich site XNis on a scheme X (assumed to
be noetherian, separated and of finite Krull dimension), and the big Nisnevich
sites Sm/BNis or Sch/BNis , as well as the Zariski versions XZar , Sm/BZar ,
etc. We describe the Nisnevich version for spectra below; the definitions and
results for the Zariski topology and for spaces are exactly parallel.
Definition 2.3.1. A map f : E → F of presheaves of spectra on XNis is a local
weak equivalence if the induced map on the Nisnevich sheaf of stable homotopy
s
s
(F )Nis is an isomorphism of sheaves for all m. A
(E)Nis → πm
groups f∗ : πm
map f : E → F of presheaves of spectra on Sm/BNis or Sch/BNis is a local
weak equivalence if the restriction of f to XNis is a local weak equivalence for
all X ∈ Sm/B or X ∈ Sch/B.
The Nisnevich local model structure on the category of presheaves of spectra
on XNis has cofibrations given pointwise, weak equivalences the local weak
equivalences and fibrations are characterized by having the RLP with respect to trivial cofibrations. We write Spt(XNis ) for this model category, and
HSpt(XNis ) for the associated homotopy category. The Nisnevich local model
categories Spt(Sm/BNis ) and Spt(Sch/BNis ), with homotopy categories
HSpt(Sm/BNis ) and HSpt(Sch/BNis ), are defined similarly. A similar localization gives model categories of presheaves of spaces Spc(XNis ), Spc(XZar ),
Spc(Sm/BNis ), etc., and homotopy categories HSpc(XNis ), HSpc(XZar ),
HSpc(Sm/BNis ), etc. We also have the adjoint pair (Σ∞ , Ω∞ ) in this setting. For details, we refer the reader to [19].
Documenta Mathematica 12 (2007) 71–146
76
Marc Levine
Remark 2.3.2. Let E be in Spt(Sm/BNis ), and let
W′
i′
/ Y′
f
W
i
/X
be an elementary Nisnevich square, i.e., f is ´etale, i : W → X is a closed
immersion, the square is cartesian, and W ′ → W is an isomorphism, with X
and X ′ in Sm/B (see [34, Definition 1.3, pg. 96]) .
If E is fibrant in Spt(Sm/BNis ) then E transforms each elementary Nisnevich
square to a homotopy cartesian square in Spt. Conversely, suppose that E
transforms each elementary Nisnevich square to a homotopy cartesian square
in Spt. Then E is quasi-fibrant, i.e., for all Y ∈ Sm/B, the canonical map
E(Y ) → Ef ib (Y ), where Ef ib is the fibrant model of E, is a weak equivalence.
See [19] for details.
If we define an elementary Zariski square as above, with X ′ → X an open
immersion, the same holds in the model category Spt(Sm/BZar ). More precisely, one can show (see e.g. [45]) that, if E transforms each elementary Zarisk
square to a homotopy cartesian square in Spt, then E satisfies Mayer-Vietoris
for the Zariski topology: if X ∈ Sm/B is a union of Zariski open subschemes
U and V , then the evident sequence
E(X) → E(U ) ⊕ E(V ) → E(U ∩ V )
is a homotopy fiber sequence in SH.
Remark 2.3.3. Let C be a small category with an initial object ∅ and admiting
finite coproducts over ∅, denoted X ∐ Y . A functor E : C op → Spt is called
additive if for each X, Y in C, the canonical map
E(X ∐ Y ) → E(X) ⊕ E(Y )
in SH is an isomorphism. It is easy to show that if E ∈ Spt(Sm/B) satisfies
Mayer-Vietoris for the Zariski topology, and E(∅) ∼
= 0 in SH, then E is additive.
From now on, we will assume that all our presheaves of spectra E satisfy
E(∅) ∼
= 0 in SH.
2.4. A1 -local structure. One can perform a Bousfield localization on
Spc(Sm/BNis ) or Spt(Sm/BNis ) so that the maps Σ∞ X × A1+ → Σ∞ X+
induced by the projections X × A1 → X become weak equivalences. We call
the resulting model structure the Nisnevich-local A1 -model structure, denoted
SpcA1 (Sm/BNis ) or SptA1 (Sm/BNis ). One has the Zariski-local versions as
well. We denote the homotopy categories for the Nisnevich version by HA1 (B)
(for spaces) and SHA1 (B) (for spectra). For the Zariski versions, we indicate
the topology in the notation. We also have the adjoint pair (Σ∞ , Ω∞ ) in this
setting. For details, see [30, 31, 34].
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
77
2.5. Additional notation. Given W ∈ Sm/S, we have restriction functors
Spc(S) → Spc(WZar )
Spt(S) → Spt(WZar );
we write the restriction of some E ∈ Spc(S) to Spc(WZar ) as E(WZar ). We
use a similar notation for the restriction of E to Spt(WZar ), or for restrictions
to WNis . More generally, if p : Y → W is a morphism in Sm/S, we write
E(Y /WZar ) for the presheaf U 7→ E(Y ×W U ) on WZar .
For Z ⊂ Y a closed subset, Y ∈ Sm/S and for E ∈ Spc(S) or E ∈ Spt(S),
we write E Z (Y ) for the homotopy fiber of the restriction map
E(Y ) → E(Y \ Z).
ZZar
We define the presheaf E
(Y ) by setting, for U ⊂ Z a Zariski open subscheme with closed complement F ,
E ZZar (Y )(U ) := E U (Y \ F ).
A co-presheaf on a category C with values in A is just an A-valued preheaf on
C op .
As usual, we let ∆n denote the algebraic n-simplex
X
∆n := Spec Z[t0 , . . . , tn ]/
ti − 1,
i
and ∆∗ the cosimplicial scheme n 7→ ∆n . For a scheme X, we have ∆nX :=
X × ∆n and the cosimplicial scheme ∆∗X .
Let B be a scheme as above. For E ∈ Spc(B) or in Spt(B), we say that E is
homotopy invariant if for all X ∈ Sm/B, the pull-back map E(X) → E(X×A1 )
is a weak equivalence (resp., stable weak equivalence). We say that E satisfies
Nisnevich excision if E transforms elementary Nisnevich squares to homotopy
cartesian squares.
3. Tubular neighborhoods for smooth pairs
Let i : W → X be a closed immersion in Sm/k. In this section, we construct the
ˆ
tubular neighborhood τǫX (W ) of W in X as a functor from WZar to cosimplicial
ˆ
pro-k-schemes. Given E ∈ Spc(k), we can evaluate E on τǫX (W ), yielding the
ˆ
X
presheaf of spaces E(τǫ (W )) on WZar , which is our main object of study.
ˆ
3.1. The cosimplicial pro-scheme τǫX (W ). For a closed immersion W → T
W
in Sm/k, let TNis
be the category of Nisnevich neighborhoods of W in T , i.e.,
objects are ´etale maps p : T ′ → T of finite type, together with a section
s : W → T ′ to p over W . Morphisms are morphisms over T which respect the
W
sections. Note that TNis
is a left-filtering essentially small category.
′
h
Sending (p : T → T, s : W → T ′ ) to T ′ ∈ Sm/k defines the pro-object TˆW
of
′
Sm/k; the sections s : W → T give rise to a map of the constant pro-scheme
h
W to TˆW
, denoted
h
ˆiW : W → TˆW
.
Documenta Mathematica 12 (2007) 71–146
78
Marc Levine
Given a k-morphism f : S → T , and closed immersions iV : V → S, iW : W →
T such that f ◦ iV factors through iW (by f¯ : V → W ), we have the pull-back
functor
W
V
f ∗ : TNis
→ SNis
,
f ∗ (T ′ → T, s : W → T ′ ) := (T ′ ×T S, (s ◦ f¯, iV )).
h
This gives us the map of pro-objects f h : SˆVh → TˆW
, so that sending W → T
h
h
to TˆW and f to f becomes a pseudo-functor.
h
We let f h : SˆVh → TˆW
denote the induced map on pro-schemes. If f happens
to be a Nisnevich neighborhood of W → X (so f¯ : V → W is an isomorphism)
h
then f h : SˆVh → TˆW
is clearly an isomorphism.
h
Remark 3.1.1. The pseudo-functor (W → T ) 7→ TˆW
can be rectified to an
W
W
honest functor by first replacing TNis with the cofinal subcategory TNis,0
of
neighborhoods T ′ → T , s : W → T ′ such that each connected component of T ′
W
has only identity
has non-empty intersection with s(W ). One notes that TNis,0
W
W
automorphisms, so we replace TNis,0 with a choice of a full subcategory TNis,00
W
giving a set of representatives of the isomorphism classes in TNis,0 . Given a
i
i
W
V
→ T ) as above,
S) → (W −−
map of pairs of closed immersions f : (V −→
∗
we modify the pull-back functor f defined above by passing to the connected
component of (s ◦ f¯, iV )(V ) in T ′ ×T S. We thus have the honest functor
W
h
(W → T ) 7→ TNis,00
which yields an equivalent pro-object TˆW
.
As pointed out by the referee, one can also achieve strict functoriality by rectifying the fiber product; in any case, we will use a strictly functorial version
from now on without comment.
n )h . The
ˆn
d
For a closed immersion i : W → X in Sm/k, set ∆
:= (∆
n
X,W
cosimplicial scheme
X ∆W
∆∗X : Ord → Sm/k
[n] 7→ ∆nX
thus gives rise to the cosimplicial pro-scheme
ˆ∗
∆
: Ord → Pro-Sm/k
X,W
ˆn
[n] 7→ ∆
X,W
n )h
d
give the closed immersion of cosimplicial
The maps ˆi∆nW : ∆nW → (∆
X ∆n
W
pro-schemes
ˆ ∗X,W .
ˆiW : ∆∗W → ∆
ˆn
Also, the canonical maps πn : ∆
→ ∆n define the map
X,W
πX,W :
ˆ∗
∆
X,W
X
→ ∆∗X .
Let (p : X ′ → X, s : W → X ′ ) be a Nisnevich neighborhood of (W, X). The
map
ˆn
ˆn′ → ∆
p:∆
X,W
X ,W
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
79
is an isomorphism respecting the closed immersions ˆiW . Thus, sending a Zariski
ˆn
open subscheme U ⊂ W with complement F ⊂ W ⊂ X to ∆
X\F,U defines a
n
ˆ
on WZar with values in pro-objects of Sm/k; we write
co-presheaf ∆
ˆ
ˆ
X,W
Zar
τǫX (W ) for the cosimplicial object
ˆ nˆ
n 7→ ∆
X,W
Zar
.
ˆ in the notation because the co-presheaf ∆
ˆn
is depends only
We use the X
ˆ
X,W
Zar
on the Nisnevich neighborhood of W in X.
Let ∆∗WZar denote the co-presheaf on WZar defined by U 7→ ∆∗U . The closed
immersions ˆiU define the natural transformation
ˆiW : ∆∗W → τǫXˆ (W ).
Zar
The maps πX\F,W \F for F ⊂ W a Zariski closed subset define the map
ˆ
πX,W : τǫX (W ) → ∆∗X|WZar
where X|WZar is the co-presheaf W \ F 7→ X \ F on WZar . We let
(3.1.1)
ˆ
π
¯X,W : τǫX (W ) → X|WZar
denote the composition of πX.W with the projection ∆∗X|WZar → X|WZar .
3.2. Evaluation on spaces. Let i : W → T be a closed immersion in Sm/k.
For E ∈ Spc(T ), we have the space E(TˆhW ), defined by
h
E(TˆW
) :=
colim
W
(p:T ′ →T,s:W →T ′ )∈TNis
E(T ′ ).
Given a Nisnevich neighborhood (p : T ′ → T, s : W → T ′ ), we have the
isomorphism
h
p∗ : E(Tˆh ) → E(Tˆ′
).
W
s(W )
Thus, for each open subscheme j : U → W , we may evaluate E on the
ˆ
cosimplicial pro-scheme τǫX (W )(U ), giving us the presheaf of simplicial spectra
ˆ
E(τǫX (W )) on WZar :
ˆ
ˆ
E(τǫX (W ))(U ) := E(τǫX (W )(U )).
ˆ
Now suppose that E is in Spc(k). The map ˆiW : ∆∗WZar → τǫX (W )) gives us
the map of presheaves on WZar
ˆ
i∗W : E(τǫX (W )) → E(∆∗WZar ).
Similarly, the map πX,W gives the map of presheaves on WZar
ˆ
∗
πX,W
: E(∆∗X|WZar ) → E(τǫX (W )).
The main result of this section is
Documenta Mathematica 12 (2007) 71–146
80
Marc Levine
ˆ
Theorem 3.2.1. Let E be in Spc(k). Then the map i∗W : E(τǫX (W )) →
E(∆∗WZar ) is a weak equivalence for the Zariski-local model structure, i.e., for
each point w ∈ W , the map i∗W,w on the stalks at w is a weak equivalence of
the associated total space.
3.3. Proof of Theorem 3.2.1. The proof relies on two lemmas.
Lemma 3.3.1. Let i : W → X be a closed immersion in Sm/k, giving the closed
immersion A1W → A1X . For t ∈ A1 (k), we have the section it : W → A1W ,
it (w) := (w, t). Then for each E ∈ Spc(k), the maps
ˆ ∗ 1 1 ) → E(∆
ˆ ∗X,W )
i∗0 , i∗1 : E(∆
AX ,AW
are homotopic.
Proof. This is just an adaptation of the standard triangulation argument. For
each order-preserving map g = (g1 , g2 ) : [m] → [1] × [n], let
Tg : ∆m → ∆1 × ∆n ,
be the affine-linear extension of the map on the vertices
vi 7→ (vg1 (i) , vg2 (i) ).
idX × Tg induces the map
ˆ m → (∆1\
Tˆg : ∆
× ∆nX )h∆1 ×∆n
X,W
W
We note that the isomorphism (t0 , t1 ) 7→ t0 of (∆1 , v1 , v0 ) with (A1 , 0, 1) induces
an isomorphism of cosimplicial schemes
ˆ ∗1
∆
A
1
X ,AW
The maps
∼
× ∆∗X )h∆1 ×∆∗ .
= (∆1\
W
ˆ n1
Tˆg∗ : E(∆
A
1
X ,AW
ˆm )
) → E(∆
X,W
induce a simplicial homotopy T between i∗0 and i∗1 . Indeed, we have the simplicial sets ∆[n] : HomOrd (−, [n]). Let (∆1 × ∆∗ )∆[1] be the cosimplicial scheme
Y
n 7→ (∆1 × ∆n )∆[1]([n]) :=
∆1 × ∆n
s∈∆[1]([n])
where the product is ×Z . The inclusions δ0 , δ1 : [0] → [1] thus induce the maps
of cosimplicial schemes
δ0∗ , δ1∗ : (∆1 ×k ∆∗ )∆[1] → ∆1 ×k ∆∗ .
The maps Tg satisfy the identities necessary to define a map of cosimplicial
schemes
T : ∆∗ → (∆1 × ∆∗ )∆[1] .
with δ0∗ ◦ T = i0 , δ1∗ ◦ T = i1 . Applying the functor h , we see that the maps Tˆg
define the map of cosimplicial schemes
ˆ ∗X,W → (∆
ˆ ∗ 1 1 )∆[1] ,
Tˆ : ∆
AX ,AW
with δ0∗ ◦ Tˆ = ˆi0 , δ1∗ ◦ Tˆ = ˆi1 ; we then apply E.
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
81
Lemma 3.3.2. Take W ∈ Smk . Let X = AnW and let i : W → X be the
ˆ ∗ ) → E(∆∗ ) is a homotopy equivalence.
0-section. Then i∗W : E(∆
X,W
W
Proof. Let p : X → W be the projection, giving the map
ˆ ∗X,W → ∆
ˆ ∗W,W = ∆∗W
pˆ : ∆
ˆ ∗ ). Clearly ˆi∗ ◦ pˆ∗ = id, so it suffices to show that
and pˆ∗ : E(∆∗W ) → E(∆
X,W
W
pˆ∗ ◦ ˆi∗W is homotopic to the identity.
For this, we use the multiplication map µ : A1 × An → An ,
µ(t; x1 , . . . , xn ) := (tx1 , . . . , txn ).
The map µ × id∆∗ induces the map
\
µ
ˆ : (A1 ×\
AnW × ∆∗ )hA1 ×0W ×∆∗ → (AnW
× ∆∗ )h0W ×∆∗
with µ
ˆ ◦ ˆi0 = ˆiW ◦ pˆ and µ
ˆ ◦ ˆi1 = id. Since ˆi∗0 and ˆi∗1 are homotopic by
Lemma 3.3.1, the proof is complete.
To complete the proof of Theorem 3.2.1, take a point w ∈ W . Then replacing
X with a Zariski open neighborhood of w, we may assume there is a Nisnevich
neigborhood X ′ → X, s : W → X ′ of W in X such that W → X ′ is in
turn a Nisnevich neighborhood of the zero-section W → AnW , n = codimX W .
ˆ n ) is thus weakly equivalent to E(∆
ˆ nn
Since E(∆
X,W
AW ,0W ), the result follows
from Lemma 3.3.2.
Corollary 3.3.3. Suppose that E ∈ Spc(Sm/k), resp. E ∈ Spt(Sm/k) is
homotopy invariant. Then for i : W → X a closed immersion, there is a
natural isomorphism in HSpc(WZar ), resp. HSpt(WZar )
ˆ
ˆ
E(τǫX (W )) ∼
= E(τǫNi (0W ))
Here Ni is the normal bundle of the immersion i, and 0W ⊂ Ni is the 0-section.
Proof. This follows directly from Theorem 3.2.1: Since E is homotopy invariant, the canonical map
E(T ) → E(∆∗T )
is a weak equivalence for each T ∈ Sm/k. The desired isomorphism in the
respective homotopy category is constructed by composing the isomorphisms
ˆ
i∗
W
E(τǫX (W )) −−
→ E(∆∗WZar ) ← E(WZar )
i∗
0
ˆ
W
= E(0W Zar ) → E(∆∗0W Zar ) ←−
− E(τǫNi (0W )).
4. Punctured tubular neighborhoods
ˆ
Our real interest is not in the tubular neighborhood τǫX (W ), but in the puncˆ
tured tubular neighborhood τǫX (W )0 . In this section, we define this object and
discuss its basic properties.
Documenta Mathematica 12 (2007) 71–146
82
Marc Levine
4.1. Definition of the punctured neighborhood. Let i : W → X be
a closed immersion in Sm/k. We have the closed immersion of cosimplicial
pro-schemes
ˆ ∗X,W
ˆi : ∆∗W → ∆
ˆn
ˆn
giving for each n the open complement ∆
:= ∆
\ ∆n . We may pass
X\W
X,W
W
to the cofinal subcategory of Nisnevich neighborhoods of ∆nW in ∆nX ,
(p : T → ∆nX , s : ∆nW → T )
for which the diagram
T \ s(∆nW )
/T
∆nX \ ∆nW
/ ∆nX
ˆn
is cartesian, giving us the cosimplicial proscheme n 7→ ∆
X\W and the map
ˆ∗
ˆ∗
ˆj : ∆
X\W → ∆X,W ,
∗
ˆ∗
ˆ∗
which defines the “open complement” ∆
X\W of ∆W in ∆X,W . Extending this
construction to all open subschemes of X, we have the co-presheaf on WZar ,
ˆ∗
U = W \ F 7→ ∆
,
(X\F )\U
ˆ
τǫX (W )0 .
which we denote by
Let ∆n(X\W )Zar be the constant co-presheaf on WZar with value ∆nX\W , giving
the cosimplicial co-presheaf ∆∗(X\W )Zar . The maps
ˆ∗
ˆ∗
ˆjU : ∆
(X\F )\U → ∆(X\F ),U
ˆ
ˆ
∗
ˆ∗
define the map ˆj : τǫX (W )0 → τǫX (W ). The maps ∆
U \W ∩U → ∆X\W give us
the map
ˆ
π : τǫX (W )0 → ∆∗X\W
where we view ∆∗X\W as the constant co-sheaf on WZar .
ˆ
To give a really useful result on the presheaf E(τǫX (W )0 ), we will need to
impose additional conditions on E. These are
(1) E is homotopy invariant
(2) E satisfies Nisnevich. excision
One important consequence of these properties is the purity theorem of MorelVoevodsky:
Theorem 4.1.1 (Purity [34, theorem 2.23]). Suppose E ∈ Spt(k) is homotopy
invariant and satisfies Nisnevich excision. Let i : W → X be a closed immersion in Sm/k and s : W → Ni the 0-section of the normal bundle. Then there
is an isomorphism in HSpt(WZar )
E WZar (X) → E WZar (Ni )
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
83
Let E(X|WZar ) be the presheaf on WZar
W \ F 7→ E(X \ F )
and E(X \ W ) the constant presheaf.
Let
ˆ
res : E(X|WZar ) → E(τǫX (W ))
ˆ
res0 : E(X \ W ) → E(τǫX (W )0 )
ˆ
ˆ
be the pull-back by the natural maps τǫX (W )(W \ F ) → X \ F , τǫX (W )0 →
∗
ˆ
X \W . Let E ∆W (τǫX (W )) ∈ Spt(WZar ) be the homotopy fiber of the restriction
map
ˆj : E(τǫXˆ (W )) → E(τǫXˆ (W )0 ).
The commutative diagram in Spt(WZar )
E(X|WZar )
j∗
/ E(X \ W )
res
ˆ
res0
E(τǫX (W ))
ˆ
j∗
/ E(τ Xˆ (W )0 )
ǫ
induces the map of homotopy fiber sequences
E WZar (X)
/ E(X \ W )
res
ψ
∗
j∗
/ E(X|WZar )
res0
ˆ
E ∆W (τǫX (W ))
/ E(τ Xˆ (W ))
ǫ
j∗
/ E(τ Xˆ (W )0 )
ǫ
ˆ
We can now state the main result for E(τǫX (W )0 ).
Theorem 4.1.2. Suppose that E ∈ Spt(k) is homotopy invariant and satisfies
Nisnevich excision. Let i : W → X be a closed immersion in Sm/k. Then the
map ψ is a Zariski local weak equivalence.
Proof. Let i∆∗ : ∆∗W → ∆∗X be the immersion id × i. For U = W \ F ⊂ W ,
ˆ
τǫX (W )0 (U ) is the cosimplicial pro-scheme with n-cosimplices
ˆ
n
ˆn
τǫX (W )0 (U )n = ∆
X\F,U \ ∆U
so by Nisnevich excision we have the natural isomorphism
∗
∗
ˆ
α : E ∆WZar (∆∗X|WZar ) → E ∆W (τǫX (W )),
∗
where E ∆WZar (∆∗X|WZar )(W \F ) is the total spectrum of the simplicial spectrum
n
n 7→ E ∆W \F (∆nX\F ).
The homotopy invariance of E implies that the pull-back
n
E W \F (X \ F ) → E ∆W \F (∆nX\F )
Documenta Mathematica 12 (2007) 71–146
84
Marc Levine
is a weak equivalence for all n, so we have the weak equivalence
∗
β : E WZar (X) → E ∆WZar (∆∗X|WZar ).
It follows from the construction that ψ = αβ, completing the proof.
Corollary 4.1.3. There is a distinguished triangle in HSpt(WZar )
ˆ
E WZar (X) → E(WZar ) → E(τǫX (W )0 )
ˆ
Proof. By Theorem 3.2.1, the map ˆi∗ : E(τǫX (W )) → E(∆∗WZar ) is a weak
equivalence; using homotopy invariance again, the map
E(WZar ) → E(∆∗WZar )
is a weak equivalence. Combining this with Theorem 4.1.2 yields the result.
For homotopy invariant E ∈ Spt(k), we let
ˆ
ˆ
φE : E(τǫNi (0W )) → E(τǫX (W )).
be the isomorphism in HSpt(WZar ) given by corollary 3.3.3.
Corollary 4.1.4. Suppose that E ∈ Spt(k) is homotopy invariant and satisfies Nisnevich excision. Let i : W → X be a closed immersion in Sm/k and
let Ni0 = Ni \ 0W .
(1) The restriction maps
ˆ
res : E(Ni /WZar ) → E(τǫNi (0W ))
ˆ
res0 : E(Ni0 /WZar ) → E(τǫNi (0W )0 )
are weak equivalences in Spt(WZar ).
(2) There is a canonical isomorphism in HSpt(WZar )
ˆ
ˆ
φ0E : E(τǫNi (0W )0 ) → E(τǫX (W )0 )
(3) Consider the diagram (in HSpt(WZar )):
E 0W Zar (Ni )
/ E(N 0 /WZar )
/ E(Ni /WZar )
i
res0E
resE
E 0W Zar (Ni )
/ E(τ Nˆi (0 ))
W
ǫ
π
E WZar (X)
jˆ
N
∗
φ0E
φE
/ E(τ Xˆ (W ))
ǫ O
ˆ
j∗
/ E(X|WZar )
/ E(τ Xˆ (W )0 )
ǫ O
res0E
resE
E WZar (X)
/ E(τ Nˆi (0 )0 )
W
ǫ
j∗
/ E(X \ W )
The first and last rows are the homotopy fiber sequences defining the presheaves
E 0W Zar (Ni ) and E WZar (X), respectively, the second row and third rows are the
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
85
distinguished triangles of Theorem 4.1.2, and π is the Morel-Voevodsky purity
isomorphism. Then this diagram commutes and each triple of vertical maps
defines a map of distinguished triangles.
Proof. It follows directly from the weak equivalence (in Theorem 4.1.2) of the
homotopy fiber of
ˆj ∗ : E(τǫXˆ (W )) → E(τǫXˆ (W )0 )
with E WZar (X) that the triple (id, resE , res0E ) defines a map of distinguished
triangles. The same holds for the map of the first row to the second row; we
now verify that this latter map is an isomorphism of distinguished triangles.
For this, let s : W → Ni be the zero-section. We have the isomorphism
ˆ
i∗W : E(τǫNi (0W )) → E(WZar ) defined as the diagram of weak equivalences
i∗
∆∗
ˆ
ι
W
0∗
E(τǫNi (0W )) −−−−Zar
−→ E(∆∗WZar ) ←−
− E(WZar ).
From this, it is easy to check that the diagram
resE
/ E(τ Nˆi (0 ))
E(Ni /WZar )
W
ǫ
OOO
OOO
OOO
i∗
W
OOO
s∗
'
E(WZar )
commutes in HSpt(WZar ). As E is homotopy invariant, s∗ is an isomorphism,
hence resE is an isomorphism as well. This completes the proof of (1).
The proof of (2) and (3) uses the standard deformation diagram. Let µ
¯ : Y¯ →
X × A1 be the blow-up of X × A1 along W , let µ
¯−1 [X × 0] denote the proper
transform, and let µ : Y → X × A1 be the open subscheme Y¯ \ µ
¯−1 [X × 0].
Let p : Y → A1 be p2 ◦ µ. Then p−1 (0) = Ni , p−1 (1) = X × 1 = X, and Y
contains the proper transform µ
¯−1 [W × A1 ], which is mapped isomorphically
1
1
by µ to W × A ⊂ X × A . We let ˜i : W × A1 → Y be the resulting closed
immersion. The restriction of ˜i to W × 0 is the zero-section s : W → Ni and
the restriction of ˜i to W × 1 is i : W → X. The resulting diagram is
(4.1.1)
W
i0
i1
/ W × A1 o
s
W
˜i
Ni
i0
/Y o
i
i1
X
p
p0
0
i0
/ A1 o
p1
i1
1
Documenta Mathematica 12 (2007) 71–146
86
Marc Levine
Together with Theorem 4.1.2, diagram (4.1.1) gives us two maps of distinguished triangles:
h
i
ˆ
ˆ
E 0W ×A1 Zar (Y ) → E(τǫY (W × A1 )) → E(τǫY (W × A1 )0 )
i∗
and
h
h
1
−→
i
ˆ
ˆ
E WZar (X) → E(τǫX (W )) → E(τǫX (W )0 )
i
ˆ
ˆ
E 0W ×A1 Zar (Y ) → E(τǫY (W × A1 )) → E(τǫY (W × A1 )0 )
i∗
0
−→
i
h
ˆ
ˆ
E 0W Zar (Ni ) → E(τǫNi (0W )) → E(τǫNi (0W )0 )
As above, we have the commutative diagram
ˆ
E(τǫY (W × A1 ))
i∗
0
i∗
W ×A1
/ E(τ Nˆi (0 ))
W
ǫ
i∗
W
E(W × A1 )
i∗
0
/ E(W ).
As E is homotopy invariant, the maps i∗W , i∗W ×A1 and i∗0 : E(W × A1 ) → E(W )
are isomorphisms, hence
ˆ
ˆ
i∗0 : E(τǫY (W × A1 )) → E(τǫNi (0W ))
is an isomorphism. Similarly,
ˆ
ˆ
i∗1 : E(τǫY (W × A1 )) → E(τǫX (W ))
is an isomorphism. The proof of the Morel-Voevodsky purity theorem [34,
Theorem 2.23] shows that
i∗0 : E 0W ×A1 Zar (Y ) → E 0W Zar (Ni )
i∗1 : E 0W ×A1 Zar (Y ) → E WZar (X)
are weak equivalences; the purity isomorphism π is by definition i∗1 ◦ (i∗0 )−1 .
Thus, both i∗0 and i∗1 define isomorphisms of distinguished triangles, and
ˆ
ˆ
i∗1 ◦ (i∗0 )−1 : E(τǫNi (0W )) → E(τǫX (W ))
is the map φE . Defining φ0E to be the isomorphism
ˆ
ˆ
i∗1 ◦ (i∗0 )−1 : E(τǫNi (0W )0 ) → E(τǫX (W )0 )
proves both (2) and (3).
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
87
Remarks 4.1.5. (1) It follows from the construction of φE and φ0E that both of
these maps are natural in E.
(2) The maps φ0E are natural in the closed immersion i : W → X in the
following sense: Suppose we have closed immersions ij : Wj → Xj , j = 1, 2
and a morphism f : (W1 , X1 ) → (W2 , X2 ) of pairs of immersions such that f
restricts to a morphism X1 \ W1 → X2 \ W2 . Fix E and let φ0jE be the map
corresponding to the immersions ij We have the evident maps
ˆ
ˆi
N
1
ˆ
ι : τǫX1 (W1 )0 → τǫX2 (W2 )0
η : τǫ
ˆi
N
2
(W1 )0 → τǫ
(W2 )0
Then the diagram
ˆi
N
2
E(τǫ
(W )0 )
φ2E
/ E(τ Yˆ (W )0 )
ǫ
η∗
ι∗
ˆi
N
1
E(τǫ
(W )0 )
φ1E
/ E(τ Xˆ (W )0 )
ǫ
commutes. Indeed, the map f induces a map of deformation diagrams.
5. The exponential map
′
If i : M → M is a submanifold of a differentiable manifold M , there is a
diffeomorphism exp of the normal bundle NM ′ /M of M ′ in M with the tubular
neighborhood τǫM (M ′ ). In addition, exp restricts to a diffeomorphism exp0 of
the punctured normal bundle NM ′ /M \ 0M ′ with the punctured tubular neighborhood τǫM (M ′ ) \ M ′ . Classically, this has been used to define the boundary map in the Gysin sequence for M ′ → M , by using the restriction map
exp0∗ : H ∗ (M \ M ′ ) → H ∗ (NM ′ /M \ 0M ′ ) followed by the Thom isomorphism
H ∗ (NM ′ /M \ 0M ′ ) ∼
= H ∗−d (M ′ ).
In this section, we use our punctured tubular neighborhood to construct a
purely algebraic version of the classical exponential map, at least for the associated suspension spectra. We will use this in section 11 to define a purely
algebraic version of the gluing of Riemann surfaces along boundary components.
5.1. Let i : W → X be a closed immersion in Sm/k with normal bundle
p : Ni → W . We have the map
exp : Ni → X
in Spc(k), defined as the composition Ni → W → X. We also have the
Morel-Voevodsky purity isomorphism
π : Th(Ni ) → X/(X \ W )
in H(k). In fact:
Documenta Mathematica 12 (2007) 71–146
88
Marc Levine
Lemma 5.1.1. The diagram
(5.1.1)
q′
Ni
/ Th(Ni )
exp
π
X
/ X/(X \ W )
q
commutes in H(k).
Proof. As we have already seen, the construction of the purity isomorphism π
relies on the deformation to the normal bundle; we retain the notation from the
proof of corollary 4.1.4. We have the total space Y → A1 of the deformation.
The fiber Y0 over 0 ∈ A1 is canonically isomorphic to Ni and the fiber Y1 over
1 is canonically isomorphic to X; the inclusions W × 0 → Y0 , W × 1 → Y1
are isomorphic to the zero-section s : W → Ni and the original inclusion
i : W → X, respectively. The proper transform µ−1 [W × A1 ] is isomorphic
to W × A1 , giving the closed immersion ι : W × A1 → Y . The diagram thus
induces maps in Spc(k):
¯i0 : Th(Ni ) → Y /(Y \ W × A1 )
¯i1 : X/X \ W → Y /(Y \ W × A1 )
which are isomorphisms in H(k) (see [34, Thm. 2.23]); the purity isomorphism
¯
is by definition π := ¯i−1
1 ◦ i0 .
We have the commutative diagram in Spc(k):
id
i0
W
O
p
/ W × A1 o
s
p1
/
W
i1
ι
i
/Y o
X
/ Y /(Y \ W × A1 ) o
X/X \ W
Ni
q′
q
Th(Ni )
¯i0
¯i1
from which the result follows directly.
Remark 5.1.2. Since we have the homotopy cofiber sequences:
Ni \ 0W → Ni → Th(Ni ) → Σ(Ni \ 0W )+
X \ W → X → X/(X \ W ) → Σ(X \ W )+
the diagram (5.1.1) induces a map
Σ(Ni \ 0W )+ → Σ(X \ W )+
in H(k), however, this map is not uniquely determined, hence is not canonical.
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
89
5.2. The construction. In this section we define a canonical map
exp0 : Σ∞ (Ni \ 0W )+ → Σ∞ (X \ W )+
in SHA1 (k) which yields the map of distinguished triangles in SHA1 (k):
/ Σ∞ Ni+
Σ∞ (Ni \ 0W )+
/ Σ∞ Th(Ni )
exp
exp0
π
/ Σ∞ X/(X \ W )
/ Σ∞ X+
Σ∞ (X \ W )+
To define exp0 , we apply Corollary 4.1.4 with E a fibrant model of Σ∞ (X \W )+ .
Denote the composition
res0
ˆ
E
E(X \ W ) −−−→
E(τǫX (W )0 )(W )
(φ0 )−1
(res0 )−1
ˆ
E
−−−
−−→ E(τǫNi (0W )0 )(W ) −−−E−−→ E(Ni0 )
by exp0∗
E . Since E is fibrant, we have canonical isomorphisms
0
, E)
π0 E(Ni0 ) ∼
= HomSHA1 (k) (Σ∞ Ni+
∞ 0
∼
= HomSHA1 (k) (Σ Ni+ , Σ∞ (X \ W )+ )
π0 E(X \ W ) ∼
= HomSHA1 (k) (Σ∞ (X \ W )+ , E)
∼ HomSH (k) (Σ∞ (X \ W )+ , Σ∞ (X \ W )+ )
=
A1
so exp0∗
E induces the map
HomSHA1 (k) (Σ∞ (X \ W )+ , Σ∞ (X \ W )+ )
exp0∗
0
, Σ∞ (X \ W )+ ).
−−−E→ HomSHA1 (k) (Σ∞ Ni+
We set
exp0 := exp0∗
E (id).
To finish the construction, we show
Proposition 5.2.1. The diagram, with rows the evident homotopy cofiber sequences,
Σ∞ (Ni \ 0W )+
/ Σ∞ Ni+
exp
exp0
Σ∞ (X \ W )+
/ Σ∞ Th(Ni )
∂
/ ΣΣ∞ (Ni \ 0W )+
Σ exp0
π
/ Σ∞ X+
/ Σ∞ X/(X \ W )
∂
/ ΣΣ∞ (X \ W )+
commutes in SHA1 (k).
Documenta Mathematica 12 (2007) 71–146
90
Marc Levine
Proof. It suffices to show that, for all fibrant E ∈ Spt(k), the diagram formed
by applying HomSHA1 (k) (−, E) to our diagram commutes. This latter diagram
is the same as applying π0 to the diagram
(5.2.1)
E(Ni \ 0W ) o
O
exp0∗
E(X \ W ) o
E 0W (Ni ) o
O
E(Ni ) o
O
exp∗
∂
ΩE(Ni \ 0W ))
O
Ω exp0∗
π∗
E W (X) o
E(X) o
ΩE(X \ W )
∂
where the rows are the evident homotopy fiber sequences. It follows by the
definition of exp0 and exp that this diagram is just the “outside” of the diagram
in Corollary 4.1.4(3), extended to make the distinguished triangles explicit.
Thus the diagram (5.2.1) commutes, which finishes the proof.
Remark 5.2.2. The exponential maps exp and exp0 are natural with respect
i′
i
to maps of closed immersions f : (W ′ −
→ X ′ ) → (W −
→ X) satisfying
cartesian condition of remark 4.1.5(2). This follows from the naturality of
isomorphisms φE , φ0E described in Remark 4.1.5, and the functoriality of
(punctured) tubular neighborhood construction.
the
the
the
6. Neighborhoods of normal crossing schemes
We extend our results to the case of a strict normal crossing divisor W ⊂ X
by using a Mayer-Vietoris construction.
6.1. Normal crossing schemes. Let D be a reduced effective Cartier divisor on a smooth k-scheme X with irreducible components D1 ,. . ., Dm . For
each I ⊂ {1, . . . , m}, we set
DI := ∩i∈I Di
We let i : D → X the inclusion. For each I 6= ∅, we let ιI : DI → D,
iI : DI → X be the inclusions; for I ⊂ J ⊂ {1, . . . , m} we have as well the
inclusion ιI,J : DJ → DI .
Recall that D is a strict normal crossing divisor if for each I, DI is smooth
over k and codimX DI = |I|.
We extend this notion a bit: We call a closed subscheme D ⊂ X a strict normal
crossing subscheme if X is in Sm/k and, locally on X, there is a smooth locally
closed subscheme Y ⊂ X containing D such that D is a strict normal crossing
divisor on Y
6.2. The tubular neighborhood. Let D ⊂ X be a strict normal crossing
subscheme with irreducible components D1 , . . . , Dm . For each I ⊂ {1, . . . , m},
ˆ
I 6= ∅, we have the tubular neighborhood co-presheaf τǫX (DI ) on DI . The
various inclusions ιI,J give us the maps of co-presheaves
ˆ
ˆ
ˆιI,J : ιI,J∗ (τǫX (DJ )) → τǫX (DI );
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
91
pushing forward by the maps ιI yields the diagram of co-presheaves on DZar
(with values in cosimplicial pro-objects of Sm/k)
ˆ
I 7→ ιI∗ (τǫX (DI ))
(6.2.1)
indexed by the non-empty I ⊂ {1, . . . , m}. We have as well the diagram of
identity co-presheaves
I 7→ ιI∗ (DIZar )
(6.2.2)
as well as the diagram
I 7→ ιI∗ (∆∗DIZar )
(6.2.3)
ˆ
We denote these diagrams by τǫX (D), D• and ∆∗D• , respectively. The projecˆ
tions pI : ∆∗DIZar → DIZar and the closed immersions ˆιDI : ∆∗DIZar → τǫX (DI )
yield the natural transformations
ˆ
ιD
p•
ˆ
•
D• ←− ∆∗D• −−→
τǫX (D).
Now take E ∈ Spt(k). Applying E to the diagram (6.2.1) yields the diagram
of presheaves on DZar
ˆ
I 7→ ιI∗ (E(τǫX (DI )))
Similarly, applying E to (6.2.2) and (6.2.3) yields the diagrams of presheaves
on DZar
I 7→ ιI∗ (E(DIZar ))
and
I 7→ ιI∗ (E(∆∗DIZar )).
Definition 6.2.1. For D ⊂ X a strict normal crossing subscheme and E ∈
Spt(k), set
ˆ
ˆ
E(τǫX (D)) := holim ιI∗ (E(τǫX (DI ))).
I6=∅
Similarly, set
E(D• ) := holim ιI∗ (E(DI ))
I6=∅
E(∆∗D• )
:= holim ιI∗ (E(∆∗DI ))
I6=∅
The natural transformations ˆιD and p• yield the maps of presheaves on DZar
p∗
ˆ
ι∗
ˆ
•
D
E(D• ) −→
E(∆∗D• ) ←−
− E(τǫX (D)).
Proposition 6.2.2. Suppose E ∈ Spt(Sm/k) is homotopy invariant and satisfies Nisnevich excision. Then the maps ˆι∗D and p∗• are Zariski-local weak
equivalences.
Documenta Mathematica 12 (2007) 71–146
92
Marc Levine
Proof. The maps p∗I are pointwise weak equivalences by homotopy invariance.
By Theorem 3.2.1, the maps ˆιDI are Zariski-local weak equivalences. Since
the homotopy limits are finite, the stalk of each homotopy limit is weakly
equivalence to the homotopy limit of the stalks. By [8] this suffices to conclude
that the map on the homotopy limits is a Zariski-local weak equivalence.
ˆ
Remark 6.2.3. One could also attempt a more direct definition of τǫX (D) by
just using our definition in the smooth case i : W → X and replacing the
smooth W with the normal crossing scheme D, in other words, the co-presheaf
on DZar
ˆ∗
D \ F 7→ ∆
X\F,D\F .
ˆ
ˆ
Labeling this choice τǫX (D)naive , and considering τǫX (D)naive as a constant diagram, we have the evident map of diagrams
ˆ
ˆ
φ : τǫX (D) → τǫX (D)naive
We were unable to determine if φ induces a weak equivalence after evaulation
on E ∈ Spt(k), even assuming that E is homotopy invariant and satisfies
Nisnevich excision. We were also unable to construct such an E for which φ
fails to be a weak equivalence.
6.3. The punctured tubular neighborhood. To define the punctured
ˆ
tubular neighborhood τǫX (D)0 , we proceed as follows: Fix a subset I ⊂
{1, . . . , m}, I 6= ∅. Let p : X ′ → X, s : DI → X ′ be a Nisnevich neighborhood
of DI in X, and let DX ′ = p−1 (D). Sending X ′ → X to ∆nDX ′ gives us the
ˆn
ˆn
ˆn
pro-scheme ∆
D⊂X,DI , and the closed immersion ∆D⊂X,DI → ∆X,DI . Varying
∗
ˆ
n, we have the cosimplicial pro-scheme ∆
D⊂X,DI , and the closed immersion
∗
∗
ˆ
ˆ
.
→∆
∆
D⊂X,DI
X,DI
Take a closed subset F ⊂ DI , and let U := DI \ F . As in the definition of the
punctured tubular neighborhood of a smooth closed subscheme in section 4.1,
we pass to the appropriate cofinal subcategory of Nisnevich neighborhoods to
ˆn
ˆn
show that the open complements ∆
\∆
for varying n form a
X\F,U
D\F ⊂X\F,U
cosimplicial pro-scheme
ˆn
ˆn
n 7→ ∆
X\F,U \ ∆D\F ⊂X\F,U .
Similarly, we set
ˆ
ˆ∗
ˆ∗
τǫX (D, DI )0 (U ) := ∆
X\F,U \ ∆D\F ⊂X\F,U .
ˆ
This forms the co-presheaf τǫX (D, DI )0 on DIZar . The open immersions
ˆn
ˆn
ˆn
ˆjI (U )n : ∆
→∆
\∆
X\F,U
D\F ⊂X\F,U
X\F,U
define the map
ˆjI (U ) : τǫXˆ (D, DI )0 (U ) → τǫXˆ (DI )(U ),
giving the map of co-presheaves
ˆjI : τǫXˆ (D, DI )0 → τǫXˆ (DI ).
Documenta Mathematica 12 (2007) 71–146
Motivic Tubular Neighborhoods
93
ˆ∗
ˆ∗
For J ⊂ I, we have the map ˆιJ,I : ∆
X,DI → ∆X,DJ and
ˆ∗
ˆ∗
ˆι−1
J,I (∆D⊂X,DJ ) = ∆D⊂X,DI .
ˆ
ˆ
Thus we have the map ˆι0J,I : τǫX (D, DI )0 → τǫX (D, DJ )0 and the diagram of
co-presheaves on DZar
ˆ
I 7→ ιI∗ (τǫX (D, DI )0 )
(6.3.1)
ˆ
which we denote by τǫX (D)0 . The maps ˆjI define the map
ˆj : τǫXˆ (D)0 → τǫXˆ (D).
ˆ
The projection maps πI : τǫX (DI ) → X (where we consider X as the constant
ˆ
co-presheaf on DIZar ) restrict to maps πI0 : τǫX (D, DI )0 → X \ D, which in turn
induce the map
ˆ
π 0 : τǫX (D)0 → X \ D,
where we consider X \ D the constant diagram of constant co-presheaves on
DZar .
ˆ
Definition 6.3.1. For E ∈ Spt(k), let E(τǫX (D)0 ) be the presheaf on DZar ,
ˆ
E(τǫX (D)0 ) :=
holim
ˆ
∅6=I⊂{1,...,m}
ιI∗ E(τǫX (D, DI )0 ).
The map ˆj defines the map of presheaves
ˆj ∗ : E(τǫXˆ (D)) → E(τǫXˆ (D)0 ).
ˆ
We let E DZar (τǫX (D)) denote the homotopy fiber of ˆj ∗ . Via the commutative
diagram
E(X \ F )
j∗
/ E(X \ D)
π∗
ˆ
E(τǫX (D))(D \ F )
π 0∗
ˆ
j∗
/ E(τ Xˆ (D))0 (D \ F )
ǫ
we have the canonical map
ˆ
∗
πD
: E DZar (X) → E DZar (τǫX (D)).
∗
We want to show that the map πD
is a weak equivalence, assuming that E is
homotopy invariant and satisfies Nisnevich excision. We first consider a simpler
situation. We begin by noting the following
Lemma 6.3.2. Let n0 denote the category of non-empty subsets of {1, . . . , n}
with maps the inclusions, let C be a small category and let F : C × n0 →
op
op
SptOrd be a functor. Let holimn0 F : C → SptOrd be the functor with
Documenta Mathematica 12 (2007) 71–146
94
Marc Levine
value the simplicial spectrum m 7→ holimn0 F (i × [m]) at i ∈ C. There is a
isomorphism
Tot(holim
F ) → holim
Tot(F ).
n
n
0
0
op
in HSpt(C ).
Proof. Letting n be the category of all subsets of {1, . . . , n} (including the
empty set), we may extend F to F ∗ : n → Spt(Ordop ) by F ∗ (∅) = 0.
Similarly, given a functor G : n → Spt, we may extend G to G♮ : n+1
→ Spt
0
by G♮ (I) = 0, G♮ (I ∪ {n + 1}) = G(I) for I ⊂ {1, . . . , n}. We define the iterated
homotopy fiber of G, fibn G ∈ Spt, by
G♮ .
hofibn (G) := holim
n+1
0
One easily checks that for a map g : A → B of spectra,